If you've decided that your mulligan situation is pretty bleak, you could essentially give up the current game for a superior position in the next one, by mulliganing to 0. Just keep drawing cards without doing anything else, and unless the opponent also mulliganed, they will be forced to show you at least seven of their cards (whether that be by playing them, discarding them, or otherwise) before you ever have to show them any of yours. Then assuming they don't just beat you down quickly, on the turn you finally draw your 8th card, scoop before having to discard. You should now have a pretty good idea what your opponent's deck is, while they have no insight whatsoever into yours, allowing you to sideboard for game 2 with asymmetric information.
Note that 103.4 does explicitly spell out that once you reach 0, it is not possible to mulligan any further.
If Karn restarts the game, the original game comes to an end entirely, and with it the key turn you're trying to measure. You may still end up in a so-called "turn 1" of the restarted game, but it's not actually the same turn, so using Karn would be completely wasteful. (The planeswalker version, anyway; the original creature version does have niches where it fits in.)
You might want to read Karn Liberated: it only puts back non-Aura cards from exile. There is absolutely no way to get a Paradox Haze trigger on the first turn of the game, restarted game or otherwise.
First off, if you're on the play turn 1, there is absolutely no way to make use of Paradox Haze or Braid of Fire: anything you can do to get them onto the battlefield must have happened after you missed the deadline for them to trigger.
Because Braid of Fire ends up being useless, Aggravated Assault is also useless, but it's an extremely dangerous card to look out for in any case: even a simple Tormod's Crypt would give you the freedom to imprint Black Lotus on Prototype Portal before copying the tokens, and that's way too good.
2^^(2^^2) is 2^^^3 (because there are three instances of 2), not 2^^^2. You really don't want to get stuck on 2^^^^(...anything...)^^^2, because that number reduces to simply 4.
One iteration of something like Elite Arcanist doesn't "add an arrow", it adds one to the number at the end of the arrows. Building off the previous example, 2^^(2^^(2^^2)) is 2^^^4, not 2^^^^2. Any expression you build using iterated ^^ operators will be simplified in terms of ^^^, exactly one arrow beyond the iterated one. To get to ^^^^, you have to be able to iterate an even larger process, one on the power of ^^^ to begin with. And so on: in order to add more arrows, you have to keep finding larger loops which can be dynamically repeated a number of times commensurate with the output value of your sub-loops, with each loop you can manage to wrap around the rest being worth just one extra arrow.
If you pile up multiple copies of that, it makes all the triggers go on the stack at once, which is unfortunate because it really inhibits its ability to affect the growth rate meaningfully. Furthermore, if the design is built around a series of "you can't be allowed to get this effect more than once per instance you recur a particular card", it's not even safe, same as a card like Rings of Brighthearth would be. But it is the most fascinating new engine part they've released in quite a while.
For any one mythic in a set there is two of any one rare, this is why mythics appear at an average of 1 in 8 packs.
This combined with the fact that they design for about 3½ times as many distinct cards at rare as they put at mythic (small sets' 35 to 10 = 3.5 exactly; large sets' 53 to 15 is a 3.533:1 ratio but has the nice result that all the cards mesh neatly into a full 11x11 sheet for the printing machines) is what yields the emergent result of "1 in 8".
Crosis's Catacombs doesn't force a land bounce: you have the option of simply sacrificing it, possibly with a tap in response so that it gets a similar profile to Archaeological Dig.
Couldn't Genesis Wave take over the Animist's Awakening slot, especially with no Izzet Guildmage to be a consideration? Grip of Chaos or some other card could go into the vacated slot.
Going by the current cards, it seems that Burst of Speed is the only card that permits any Mimic Vat tokens to be created at all.
Mimic Vat's ability only creates a token if the originally exiled card is still in exile as the ability resolves, without having changed zones in the mean time (the Mimic Vat, of course, can change zones or disappear without affecting anything).
World at War as the duplicate spell looks problematic, because it automatically exiles itself on resolution, without having to pay green mana to Bearscape to pull it out.
Could we arrange something like: start a single Chancellor of the Tangle, add Mana Confluence and Mana Crypt to get up to 4 mana, play some free sorcery like Land Grant, and take advantage of the existing Spellshift to turn it into something expensive, perhaps Warp World? Certainly Chaos Warp is no good because it has targets, but a resolving Warp World will indiscriminately remove all tokens and indeed all progress that isn't currently on the stack (unlike Obliterate, which can spare enchantments and planeswalkers). Still, getting a refresh on every permanent in the deck may yet be able to add to a growing repertoire of lingering triggers while still getting back the Warp World.
This is where Cultural Exchange over Scrambleverse could come in handy. It can only exchange control of creatures, so giving away Chandra would be impossible. There just needs to be some way of giving away an initial batch of creatures by other means, possibly with the dummy sorcery slot.
Every win ends up with an ultimate threat that the loser is powerless to do anything about, because if they had the resources to successfully stop it, they would. This is not unique to any "control" matchups--in a two-player rivalrous game, any increase in your own Win Probability comes with a direct and exactly commensurate decrease in your opponent's, and the only way to improve to a 100% state is by reducing the opponent to a 0% state. All games must, after all, come to a conclusion of some kind, and the only other alternative is a draw.
Since there's actually a sizable amount of breathing room, I was going to suggest Grip of Chaos to nest Psychic Battles inside it, but of course that is a red card, and it is way too easy to bounce.
Speaking of red cards, World at War and Scrambleverse both look problematic. In particular, a Spellweaver Helix imprint of Show and Tell + Scrambleverse provides easy untaps for everything in addition to the lifegain trigger. If Donate is an unacceptable substitute for Scrambleverse due to the targeting, there's always Cultural Exchange as an option, combined with something like Hunted Phantasm that can guarantee large numbers of tokens for the opponent.
Would we need something with indestructible and no other intrusive characteristics, like Darksteel Myr, to be a "holder" for the Splinter Twin while the world blows up, or can Gideon somehow be salvaged to become that holder? I don't see any way of getting back the Splinter Twin short of a Selvala's Enforcer card draw, which is good because getting Splinter Twin back would be able to trigger Staff of the Flame Magus.
On the subject of colored mana -> exiling cards from graveyards, there is Bearscape which produces relatively harmless Bear tokens. It does need a second card in the graveyard each time, somehow.
Note that 103.4 does explicitly spell out that once you reach 0, it is not possible to mulligan any further.
Because Braid of Fire ends up being useless, Aggravated Assault is also useless, but it's an extremely dangerous card to look out for in any case: even a simple Tormod's Crypt would give you the freedom to imprint Black Lotus on Prototype Portal before copying the tokens, and that's way too good.
2^^(2^^2) is 2^^^3 (because there are three instances of 2), not 2^^^2. You really don't want to get stuck on 2^^^^(...anything...)^^^2, because that number reduces to simply 4.
One iteration of something like Elite Arcanist doesn't "add an arrow", it adds one to the number at the end of the arrows. Building off the previous example, 2^^(2^^(2^^2)) is 2^^^4, not 2^^^^2. Any expression you build using iterated ^^ operators will be simplified in terms of ^^^, exactly one arrow beyond the iterated one. To get to ^^^^, you have to be able to iterate an even larger process, one on the power of ^^^ to begin with. And so on: in order to add more arrows, you have to keep finding larger loops which can be dynamically repeated a number of times commensurate with the output value of your sub-loops, with each loop you can manage to wrap around the rest being worth just one extra arrow.
If you pile up multiple copies of that, it makes all the triggers go on the stack at once, which is unfortunate because it really inhibits its ability to affect the growth rate meaningfully. Furthermore, if the design is built around a series of "you can't be allowed to get this effect more than once per instance you recur a particular card", it's not even safe, same as a card like Rings of Brighthearth would be. But it is the most fascinating new engine part they've released in quite a while.
This combined with the fact that they design for about 3½ times as many distinct cards at rare as they put at mythic (small sets' 35 to 10 = 3.5 exactly; large sets' 53 to 15 is a 3.533:1 ratio but has the nice result that all the cards mesh neatly into a full 11x11 sheet for the printing machines) is what yields the emergent result of "1 in 8".
Mimic Vat's ability only creates a token if the originally exiled card is still in exile as the ability resolves, without having changed zones in the mean time (the Mimic Vat, of course, can change zones or disappear without affecting anything).
World at War as the duplicate spell looks problematic, because it automatically exiles itself on resolution, without having to pay green mana to Bearscape to pull it out.
Speaking of red cards, World at War and Scrambleverse both look problematic. In particular, a Spellweaver Helix imprint of Show and Tell + Scrambleverse provides easy untaps for everything in addition to the lifegain trigger. If Donate is an unacceptable substitute for Scrambleverse due to the targeting, there's always Cultural Exchange as an option, combined with something like Hunted Phantasm that can guarantee large numbers of tokens for the opponent.
On the subject of colored mana -> exiling cards from graveyards, there is Bearscape which produces relatively harmless Bear tokens. It does need a second card in the graveyard each time, somehow.