Do the effects of critical hits count against shields? What if the shield that is lost due to a critical hit is regenerated (with r2-d2)? Do the effects of the critical hit still count? How do you keep track of a critical hit if its shield is regenerated?

Criticals don't count against shields. A crit means that the damage card is placed face up and the effect is resolved, since when a shield is damaged you remove a shield token instead of takeing a damage card there's no card to flip.

Of note however, wave three will be introducing at least one secondary weapon that bypasses shields entirely and deals a crit to the hull not matter what.

Oh ok, so any critical hits against a ship with shields only counts as a hit until shields are removed?

Yup. It's one of the big advantage rebels have at the moment.

Damage cards, whether face-down or face-up, are not dealt to a ship until all shields are removed (for now; see Vonpenguin's comment). If shields are remaining, an uncancelled [hit] or [crit] die result only removes a shield token. Regenerating a shield token (via R2-D2 or other ability) does not remove any face-up or face-down cards that have already been dealt, and any effects they have remain.