No and no. In both cases, you are casting everything at (essentially) the same time. You do choose the order you cast them in but they all go onto the stack before any resolve. Since Doublecast has not resolved yet, it won't be looking for the next spell cast. And, once it does resolve, Shock was already cast a long time ago so it still won't see it.
TI can't see any reason why you couldn't copy the shock.
Because you cast them all before any resolve; that ruling does not say otherwise.
Your main phase begins.
You get priority and cast Epic Experiment. You choose X=10 and pay the cost using obtained in an unspecified way.
You get priority and pass.
Opponent gets priority and passes.
The top object on the stack (the spell cast in step 2) resolves. We’ll say the 10 cards exiled are Shock, Doublecast, and 8 ineligible cards. A) You cast Shock, choosing your opponent (let’s say) as the target. B) You cast Doublecast. Then Epic Experiment goes to your graveyard.
You get priority and pass.
Opponent gets priority and passes.
The top object on the stack (the spell cast in step 5B) resolves. You now have a delayed trigger: “When you cast your next instant or sorcery spell this turn, copy that spell. You may choose new targets for the copy.” Doublecast goes to your graveyard.
You get priority and pass.
Opponent gets priority and passes.
The top object on the stack (the spell cast in step 5A) resolves. Shock deals 2 damage to your opponent and goes to your graveyard.
You get priority and ....
Yes, as the ruling said, 5A is solidly before 5B. But 8 is after 5A (or 11 after 5B if you cast them in the other order).
Private Mod Note
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Why bother with mere rulings when so many answers can be found in the Rules?
The first ruling given for Epic Experiment at the time of this writing is more relevant when an effect cares about spells—
that were cast before a given spell would be (such as in Mana Maze and storm [C.R. 702.40a] — see also this thread and this thread), or
that would be cast while a given spell is on the stack (as in split second [C.R. 702.61a] — see also this thread),
rather than when an effect cares about the next spell (of a certain kind) a player would cast after the effect begins (as in Howl of the Horde and Doublecast, as well as in Quicken). 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).
EDIT (Jun. 8, 2021): Edited, including because some rules were renumbered in the meantime.
Yes, as the ruling said, 5A is solidly before 5B. But 8 is after 5A (or 11 after 5B if you cast them in the other order).
EDIT (Jun. 8, 2021): Edited, including because some rules were renumbered in the meantime.