Star-Crossed

What happens if my opponent plays Flash on their Akali, Silent that's currently in combat after I play Star-Crossed targeting her?

Star-Crossed will resolve to no effect.

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 since they 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.

In short, the linked instructions rules are too expansive currently on what qualifies as an "ignored" instruction. To clarify this definition: an ignored instruction is one that wasn't performed as part of the execution of a spell or ability's instructions because it mistargeted (i.e. because the targeted object wasn't a legal target on resolution). This is a case where, for example, you play Hidden Blade choosing an enemy unit and they play Star-Crossed in reaction, returning their unit to hand. When Hidden Blade resolves, the unit isn't on the board, so we ignore the "Its controller draws 2." instruction. Alternatively, if Hidden Blade is given the Repeat keyword by the Academy and both executions target the same unit, this is the reason why the unit's controller doesn't draw 4 cards.

758.2.a. This will also occur if the Game Object is no longer among the set of objects that are made untargetable by a Game Effect. Example: Alpha Wildclaw and Vilemaw are controlled by the same player and located at a battlefield. An opponent targets Vilemaw with Rebuke. Before Rebuke resolves, Vilemaw's controller plays Discipline targeting Alpha Wildclaw, making Vilemaw no longer a legal target. Rebuke's controller then targets Alpha Wildclaw with Star-Crossed. When Star-Crossed resolves, Alpha Wildclaw's passive ability no longer applies to Vilemaw, making it a legal target for Rebuke. After all spells and abilities have resolved, both Vilemaw and Alpha Wildclaw will be returned to hand.