Flash

They will not.

Elder Dragon’s play effect has a number of targeting restrictions equal to the number of locations on the board. Its controller chooses up to one enemy unit at each of those locations to target with the ability and each of those units must follow their respective targeting restriction. It is the equivalent to saying “Deal 1 to up to one enemy unit at battlefield A. Deal 1 to up to one enemy unit at battlefield B. Deal 1 to up to one enemy unit in your base. Deal 1 to up to one enemy unit in Opponent A’s Base. …” Each of those targeting restrictions is fulfilled separately for each unit targeted.

If you play Flash in reaction to the play effect of Elder Dragon, any units that are moved to a location other than the one where they were targeted will no longer fulfill the targeting restriction of the Elder Dragon play effect and will be unaffected by the ability as it resolves.

No, they cannot play Repulse targeting your Alpha Strike. (Except if both of the other chosen units have changed to a non-board zone, such as via the effect of something like Grim Apothecary’s play effect, Heedless Resurrection’s additional cost, or the instructions of Thrill of the Hunt.)

A spell that targets a game object creates a targeting relationship between it and the targeted game object. As long as that targeted game object remains on the board, that relationship is maintained even if the target no longer fulfills the targeting restrictions of the spell. Even if the units in this example aren’t at a battlefield anymore, they are still being chosen by Alpha Strike. We know this is true because the spell remembers that it is targeting those units if they happen to fulfill its targeting restrictions again per rule 359.3.e.3. The only exception to this principle is if the units change to a non-board zone—in such a case, they are no longer the same game object and any relationship between them and the spell that targets them is severed. Even if they returned to the same battlefield, they won’t be legal targets and the spell won’t be choosing them.

It can help to visualize Alpha Strike’s targets with a colored arrow pointing from Alpha Strike to those units. Even if the units move to base and are no longer legal targets, they are still being chosen by Alpha Strike; we can imagine that the arrow is greyed out, but still present. Meanwhile, if they moved to a non-board zone the arrow would disappear entirely, since the game object it is connected to no longer exists. Repulse cares about the number of those arrows that point to friendly units, not the color of the arrows.

Akali will be unaffected by Star Crossed as it resolves.

Akali, Silent is what the rules refer to as “untargetable” for enemy spells and abilities when she isn’t in combat; meaning she is not a legal target for them. Enemy spells and abilities being played when she’s untargetable can’t choose her. If she becomes untargetable after she’s already been chosen by an effect and retains it as that effect begins to resolve, it will “mistarget” her. She will be unaffected by any of the spell or ability’s instructions. If an instruction chooses only her, that instruction will be ignored.

Relevant Rules: 757.1., 758., 758.1.