The latest version of Niv-Mizzet, Parun has inspired me to rebuild my old Niv deck that ran Eye of the Storm. And the brand new crazy Ravnica Storm themed enchantment, Thousand-Year Storm happens to interact with it in some interesting ways as well! So, that brings me to my second question:
When a split card is cast, Eye of the Storm's ability will exile that card. If that card is then copied, the player copying it may cast either half of that card. (If that card has fuse, fuse doesn't let that player cast both halves of that card since it's not in that player's hand.) This doesn't make Eye of the Storm's ability trigger since a card isn't cast this way, but a copy of a card is.
Unlike Eye of the Storm's ability, Thousand-Year Storm's ability will trigger whenever "you cast an instant or sorcery spell", not just an instant or sorcery card. See also this thread.
If you control Eye of the Storm and Thousand-Year Storm, here is how this situation can play out, assuming you've cast no other instant or sorcery spells this turn and no cards are exiled with Eye of the Storm yet:
You cast a Lightning Bolt card from your hand (Lightning Bolt I).
Eye of the Storm's and Thousand-Year Storm's abilities trigger. You order these abilities by putting the Eye of the Storm ability on the stack first.
Retaining priority, you cast a Brainstorm card from your hand (Brainstorm I).
Eye of the Storm's and Thousand-Year Storm's abilities trigger. You order these abilities by putting the Eye of the Storm ability on the stack first.
All players pass, then the Thousand-Year Storm ability (for Brainstorm I) resolves. You copy Brainstorm I once since you've cast one other instant or sorcery spell before it (Lightning Bolt) this turn.
All players pass, then the Eye of the Storm ability resolves. You exile Brainstorm I, then copy Brainstorm I and choose to cast the copy (Brainstorm II). This will make the Thousand-Year Storm ability trigger; that ability goes on the stack.
All players pass, then the Thousand-Year Storm ability (for Brainstorm II) resolves. You copy Brainstorm II twice since you've cast two other instant or sorcery spells before it (Lightning Bolt and Brainstorm I) this turn.
All players pass, then a copy of Brainstorm II resolves. This step is repeated until no copy of Brainstorm II remains on the stack.
All players pass, then Brainstorm II resolves.
All players pass, then the copy of Brainstorm I resolves.
All players pass, then Brainstorm I resolves.
All players pass, then the Thousand-Year Storm ability (for Lightning Bolt I) resolves. You don't copy Lightning Bolt I since you've cast no other instant or sorcery spells before it this turn.
All players pass, then the Eye of the Storm ability resolves. You exile Lightning Bolt I, then copy Lightning Bolt I and Brainstorm I and choose to cast the Brainstorm I copy and the Lightning Bolt I copy, in that order (Brainstorm III and Lightning Bolt II, respectively). This will make the Thousand-Year Storm ability trigger twice; those abilities go on the stack in the order of your choice. Assume that you order them so that the ability referring to Lightning Bolt II goes on the stack below that referring to Brainstorm III.
All players pass, then the Thousand-Year Storm ability (for Brainstorm III) resolves. You copy Brainstorm III three times since you've cast three other instant or sorcery spells before it (Lightning Bolt I and Brainstorm I and II) this turn. (Note that if an effect would have a player cast more than one spell at once, that player nevertheless casts them one at a time, since casting a spell is a multistep process [C.R. 601.2]. See also this thread.)
All players pass, then a copy of Brainstorm III resolves. This step is repeated until no copy of Brainstorm III remains on the stack.
All players pass, then the Thousand-Year Storm ability (for Lightning Bolt II) resolves. You copy Lightning Bolt II four times since you've cast four other instant or sorcery spells before it (Lightning Bolt I and Brainstorm I, II, and III) this turn.
All players pass, then a copy of Lightning Bolt II resolves. This step is repeated until no copy of Lightning Bolt II remains on the stack.
All players pass, then Lightning Bolt II resolves.
All players pass, then Brainstorm III resolves.
All players pass, then Lightning Bolt I resolves.
As a result, you can have resolve either—
five Lightning Bolt spells and ten Brainstorm spells (a possibility not shown above), or
six Lightning Bolt spells and nine Brainstorm spells.
This scenario assumes that a target for each Lightning Bolt spell can be chosen when it's cast.
1. When you cast a normal or fuse split card with Eye of the Storm, you get to choose which side of the card gets cast, though you can't fuse since that only works from hand because Eye casts from exile. For Aftermath cards, you can only cast the normal, "right side up" half of the card because Aftermath stipulates that the sideways half can only be cast from the graveyard.
2. With both enchantments on the battlefield, you casting a card (not a copy) will trigger both enchantments, and, since you control both, you get to choose the order in which they go on the stack. Either you may first have Thousand-Year Storm copy the spell for each spell cast previously, or you may first have Eye exile the spell and cast all spells exiled with it. Either way, the number of copies generated by the first Storm trigger doesn't change. The copies cast by Eye will each trigger Storm, and the later spells cast will be copied more times. Let's look at your example:
1. You cast Lightning Bolt. Eye and Storm trigger, though Storm will do nothing due to not having previously cast spells (Storm count: 1) Stack: bolt, eye
2. Eye resolves, exiling bolt from the stack and casting a copy of bolt. Storm triggers. (Storm count: 2) Stack: copy bolt, storm
3. Storm resolves, putting a copy of bolt onto the stack. (Storm count: 2) Stack: copy bolt, copy bolt
4. Presumably, you let the bolts resolve.
Now, it gets interesting.
5. You cast Brainstorm from hand. Eye and Storm trigger. (Storm count: 3) Stack: brain, Eye*, Storm*
7. Eye* resolves, exiling the original brain and casting bolt* and brain*. Storm triggers. (Storm count: 4 for the first spell and 5 for the second) Stack: brain copy*, bolt copy*, Storm 1, Storm 2
The latest version of Niv-Mizzet, Parun has inspired me to rebuild my old Niv deck that ran Eye of the Storm. And the brand new crazy Ravnica Storm themed enchantment, Thousand-Year Storm happens to interact with it in some interesting ways as well! So, that brings me to my second question:
What exactly happens when both Eye of the Storm and Thousand-Year Storm are on the battlefield and I cast Lightning Bolt? What happens if after that I cast a Brainstorm?
Thanks in advance, this is a highly convoluted interaction.
Unlike Eye of the Storm's ability, Thousand-Year Storm's ability will trigger whenever "you cast an instant or sorcery spell", not just an instant or sorcery card. See also this thread.
If you control Eye of the Storm and Thousand-Year Storm, here is how this situation can play out, assuming you've cast no other instant or sorcery spells this turn and no cards are exiled with Eye of the Storm yet:
EDIT (Jun. 6): Added assumption.
EDIT (Mar. 16, 2021): Correction.
2. With both enchantments on the battlefield, you casting a card (not a copy) will trigger both enchantments, and, since you control both, you get to choose the order in which they go on the stack. Either you may first have Thousand-Year Storm copy the spell for each spell cast previously, or you may first have Eye exile the spell and cast all spells exiled with it. Either way, the number of copies generated by the first Storm trigger doesn't change. The copies cast by Eye will each trigger Storm, and the later spells cast will be copied more times. Let's look at your example:
1. You cast Lightning Bolt. Eye and Storm trigger, though Storm will do nothing due to not having previously cast spells (Storm count: 1) Stack: bolt, eye
2. Eye resolves, exiling bolt from the stack and casting a copy of bolt. Storm triggers. (Storm count: 2) Stack: copy bolt, storm
3. Storm resolves, putting a copy of bolt onto the stack. (Storm count: 2) Stack: copy bolt, copy bolt
4. Presumably, you let the bolts resolve.
Now, it gets interesting.
5. You cast Brainstorm from hand. Eye and Storm trigger. (Storm count: 3) Stack: brain, Eye*, Storm*
6. Storm* resolves, creating 2 copies of brain. (Storm count: 3) Stack: brain, Eye*
7. Eye* resolves, exiling the original brain and casting bolt* and brain*. Storm triggers. (Storm count: 4 for the first spell and 5 for the second) Stack: brain copy*, bolt copy*, Storm 1, Storm 2
8. Storm 2 resolves, creating 4 copies of bolt* Stack: brain copy*, bolt copy*, Storm 1, 4x bolt copy
9. Bolt copies resolve Stack: brain copy*, bolt copy*, Storm 1
10. Storm 1 resolves, creating 3 copies of brain* Stack: brain copy*, bolt copy*, 3x brain copy
11. Everything else resolves
*: Order and spell may vary