No, you can't choose which animal you get from an Egg, hatching is always random, regardless of which Egg type or rarity you buy.

Mutated Eggs are worth prioritizing because Diamond and Gold mutations produce noticeably better animals than their base versions, though they only appear during periodic meteor showers rather than being available all the time.

No, you generally can't still get event Eggs like the Turkey Egg after the event ends, since they're tied to that specific event's shop and disappear once it closes.

Yes, you keep the Eggs, Animals, and Potions included with my purchase, exactly as shown in the listing.