Past in Flames checks which cards are "instant and sorcery card[s] in your graveyard" when it resolves (C.R. 608.2h). The set of cards the spell affects won't change thereafter, because the spell grants an ability to those cards (C.R. 611.2c).
On the case of Niv-Mizzet and jump-start (C.R. 702.133a), see also the following, which relies on the combination of C.R. 715.3a and 601.3e just as the case of Lier and adventurer cards does:
If an object that entered its owner's graveyard has Ulamog's last ability (in that graveyard), that ability will trigger from the graveyard (C.R. 603.2). See the following:
Where a spell or ability (such as Body of Research) says "[C]reate [a token]. Put X +1/+1 counters on it...", the it means all tokens created by the first instruction mentioned here. Even if more tokens or different tokens than otherwise are somehow created this way, X +1/+1 counters are put on each of those tokens (see also C.R. 614.6).
Here, where Beastmaster Ascension's second ability says "Beastmaster Ascension", it means only the permanent where that text appears, and not any other object even if it has the same name (C.R. 201.5a). Likewise, the text "Beastmaster Ascension" on Beastmaster Ascension's first ability means that permanent's source (see also C.R. 201.5a).
Freyalise's Winds's second ability is a static ability that works only while Freyalise's Winds is on the battlefield (C.R. 604.1, 611.3a). It replaces what happens if a permanent of a certain kind would untap during its controller's untap step; it doesn't affect whether the permanent can untap at any other time.
Some kinds of counters have game mechanics inherent to them (e.g., stun counters under C.R. 122.1d), but wind counters are not among them.
C.R. 701.19a says: "To search for a card in a zone, look at all cards in that zone (even if it’s a hidden zone) and find a card that matches the given description". The rule doesn't speak of reordering any cards in the given zone, and a library's cards normally can't be reordered (C.R. 401.2). However, it is true that searching a library is normally followed by shuffling that library, and in that case, there is little harm in reordering the library for convenience before the library is shuffled this way; indeed, shuffling a library means "randomiz[ing] the cards within it so that no player knows their order" (C.R. 701.20a).
When Blood Lust resolves, it takes into account what toughness Tetravus has, taking into account all effects that changed that toughness, such as +1/+1 counters (C.R. 613.1, 608.2h, 613.1g, 613.4c). For example, if Tetravus is 1/1 with three +1/+1 counters on it, its toughness is normally 4.
Now, if the toughness found this way is—
5 or greater, Tetravus gets +4/-4 until end of turn.
4 or less, Tetravus gets +4/-X until end of turn, where X is that toughness minus 1 (so, in the example of toughness 4, it gets +4/-3 until end of turn).
This change to power and toughness is determined only once and is not adjusted thereafter (C.R. 608.2h).
I assume you haven't cast any spells yet this turn before the beginning of this scenario.
Then, if you cast Vrestin here, The Eighth Doctor will grant it "If this permanent ... anywhere else" and The Sixth Doctor's ability will trigger (C.R. 603.2). Moreover, The Sixth Doctor's ability will get to resolve even if The Sixth Doctor leaves the battlefield in the meantime (C.R. 113.7a). Indeed, it's not by removing The Sixth Doctor from the battlefield, but by countering the ability, that the ability can be kept from resolving (C.R. 701.5a, 608.2b, 109.5).
In general, while a player casts a spell or activates an ability, no player has priority, so no other player can cast spells or activate abilities (including those that would remove permanents from the battlefield) (C.R. 601.2a-i and 602.2a-b have no priority window until the spell is cast or ability activated; C.R. 117.1a-b). In general, after a player announces a spell, the next time a player can get priority is only after that spell finishes being cast (C.R. 601.2i, 117.1a-b).
By the way, although the original Vrestin will gain "If this permanent ... anywhere else" from The Eighth Doctor, a copy of Vrestin will not necessarily have that ability, since the effect granting Vrestin the ability is not copiable (C.R. 707.2).
A card can leave a graveyard at most once, even if Brenard's ability triggers more than once or is copied, and even if multiple abilities trigger upon the same creature dying. See C.R. 400.7, 113.6m.
In normal cases, Noxious Gearhulk's activated ability can be copied with Radiant Performer for a creature with protection from red or from white, and the copy made to target that creature, since Noxious Gearhulk is neither red nor white (C.R. 702.16b). For purposes of Radiant Performer and protection abilities, a copy of a spell or ability has the same source as the original spell or ability (C.R. 707.10b). Note that, in general, abilities have no color (C.R. 603.3, 602.2a).
On the other hand, the rules seem not to clarify enough who the copy's default controller is for purposes of your question (under C.R. 707.10, a "copy of a[n] ... ability is controlled by the player under whose control it was put on the stack", but nothing in the rules appears to state under whose control a copy of an ability — as opposed to a spell — is put onto the stack by default).
See also what appears to be a cross-post of yours:
Suppose True-Name Nemesis has protection from Knight of Meadowgrain's controller.
Again, Knight of Meadowgrain's damage to True-Name Nemesis is prevented, not that it can't be dealt, but since the damage is prevented, it isn't dealt, so Knight of Meadowgrain's controller doesn't gain life equal to the prevented damage (C.R. 615.6, 120.3f, 702.16k).
Protection abilities (C.R. 702.16) don't change how combat damage is assigned (as trample does, for example [C.R. 702.19]); effects that would affect the damage a creature would _deal_, as with protection abilities, don't change the amount of combat damage creatures assign (C.R. 510.1c-d apply generally and C.R. 702.19b applies to trample). Therefore, creatures assign combat damage without regard to those effects.
Notably, creatures even assign combat damage without regard to effects that provide that certain damage can't be dealt (and without regard to effects that prevent damage).
On the case of Niv-Mizzet and jump-start (C.R. 702.133a), see also the following, which relies on the combination of C.R. 715.3a and 601.3e just as the case of Lier and adventurer cards does:
- https://www.reddit.com/r/mtgrules/comments/161to3w/question_about_niv_mizzet_and_adventures/
- https://www.mtgsalvation.com/forums/magic-fundamentals/magic-rulings/811222-og-eldrazi-shuffle-triggers
This applies to replacement effects of the kind found in Adrix and Nev as it does to those in Doubling Season's or Parallel Lives's first abilities, or in the second ability of Chatterfang, Squirrel General, among other things.
See also the following:
- https://www.reddit.com/r/mtgrules/comments/19bf2mu/slime_against_humanity_and_its_interaction_with/
- https://www.reddit.com/r/mtgrules/comments/yob1cu/when_you_create_a_token_that_has_automatic/
- https://www.reddit.com/r/mtgrules/comments/pm07m4/esix_fractals_and_counters/
- https://www.reddit.com/r/mtgrules/comments/vk0sab/adrix_and_nev_twincasters_does_the_copied_token/
- https://www.reddit.com/r/mtgrules/comments/obumk2/chatterfang_and_fractals/
- https://www.reddit.com/r/mtgrules/comments/xf9g39/ezuris_predation_and_doubling_effects/
Some kinds of counters have game mechanics inherent to them (e.g., stun counters under C.R. 122.1d), but wind counters are not among them.
Compare Freyalise's Winds with Aven Mimeomancer, Xathrid Gorgon, or Musician (C.R. 108.1).
See also:
- https://www.reddit.com/r/mtgrules/comments/rt7wfw/if_a_bounty_counter_from_mathas_fiend_seeker_is/
- https://www.reddit.com/r/mtgrules/comments/rtswzq/question_about_croak_counters/
- https://www.reddit.com/r/mtgrules/comments/u2wr4j/aquitects_will_and_unstable_frontier/
- https://www.reddit.com/r/mtgrules/comments/y2ilqa/toxrill_1_counters/
Damage marked on a creature doesn't reduce that creature's toughness or affect its power.
In general, dealing 3 damage to a creature merely marks 3 damage on that creature, without affecting the creature's toughness (C.R. 120.3e).
Thus, in your example, Rumbling Baloth's power and toughness remain 4/4 despite being dealt damage by Lightning Bolt.
Now, if the toughness found this way is—
This change to power and toughness is determined only once and is not adjusted thereafter (C.R. 608.2h).
https://www.mtgsalvation.com/forums/magic-fundamentals/magic-rulings/831025-gomazoa
Then, if you cast Vrestin here, The Eighth Doctor will grant it "If this permanent ... anywhere else" and The Sixth Doctor's ability will trigger (C.R. 603.2). Moreover, The Sixth Doctor's ability will get to resolve even if The Sixth Doctor leaves the battlefield in the meantime (C.R. 113.7a). Indeed, it's not by removing The Sixth Doctor from the battlefield, but by countering the ability, that the ability can be kept from resolving (C.R. 701.5a, 608.2b, 109.5).
In general, while a player casts a spell or activates an ability, no player has priority, so no other player can cast spells or activate abilities (including those that would remove permanents from the battlefield) (C.R. 601.2a-i and 602.2a-b have no priority window until the spell is cast or ability activated; C.R. 117.1a-b). In general, after a player announces a spell, the next time a player can get priority is only after that spell finishes being cast (C.R. 601.2i, 117.1a-b).
By the way, although the original Vrestin will gain "If this permanent ... anywhere else" from The Eighth Doctor, a copy of Vrestin will not necessarily have that ability, since the effect granting Vrestin the ability is not copiable (C.R. 707.2).
See also:
- https://www.reddit.com/r/mtgrules/comments/13l2ngv/kinzu_and_copying_triggers/
- https://www.reddit.com/r/mtgrules/comments/zxf23f/neera_wild_mage_rules_question/
- https://www.reddit.com/r/mtgrules/comments/14vwnwb/teysa_karlov_doubling_myrkul_trigger/
- https://www.reddit.com/r/mtgrules/comments/16axos2/bloodghast_and_multiple_landfall_triggers/
On the other hand, the rules seem not to clarify enough who the copy's default controller is for purposes of your question (under C.R. 707.10, a "copy of a[n] ... ability is controlled by the player under whose control it was put on the stack", but nothing in the rules appears to state under whose control a copy of an ability — as opposed to a spell — is put onto the stack by default).
See also what appears to be a cross-post of yours:
- https://www.reddit.com/r/mtgrules/comments/17ec4ak/need_help/
EDIT: Added "as opposed to a spell" after comment 4 was posted; later, added "and protection abilities".
- https://www.reddit.com/r/mtgrules/comments/ob0efx/rules_question_samut_ultimate_and_vorinclex/
- https://www.reddit.com/r/mtgrules/comments/xxqi95/casting_explore_the_underdark_with_the_baldurs/
Again, Knight of Meadowgrain's damage to True-Name Nemesis is prevented, not that it can't be dealt, but since the damage is prevented, it isn't dealt, so Knight of Meadowgrain's controller doesn't gain life equal to the prevented damage (C.R. 615.6, 120.3f, 702.16k).
Notably, creatures even assign combat damage without regard to effects that provide that certain damage can't be dealt (and without regard to effects that prevent damage).
See also:
- https://www.reddit.com/r/mtgrules/comments/prc5ap/protection_from_all_colors_vs_double_strike/
- https://www.reddit.com/r/mtgrules/comments/m569pq/dealing_double_damage_and_creatures_with_trample/ ([[Calamity Bearer]])
- https://www.reddit.com/r/mtgrules/comments/yxb0hr/calamity_bearer_ruling/ (same)
- https://www.reddit.com/r/mtgrules/comments/n72xri/torbran_calamity_bearer_effect_order/ (Torbran)
- https://www.reddit.com/r/mtgrules/comments/os3d2x/jared_carthalion_true_heir_and_trample/ ([[Jared Carthalion, True Heir]])
- https://www.reddit.com/r/mtgrules/comments/lyi93s/vorinclex_and_putrefax/ ([[Vorinclex, Monstrous Raider]]).
- https://www.reddit.com/r/mtgrules/comments/ytiwf3/how_does_blocking_work_with_a_creature_with/
- https://www.reddit.com/r/mtgrules/comments/pavg25/spirit_of_resistance_prevent_all_damage_to_you/