Can Coastal Wizard bounce multiple other creatures?
Yes, that combo works.
For the first part of your answer, whether you can activate Coastal Wizard multiple times, yes you can:
Whenever you, as the active player, cast a spell/activate an ability, or whenever a spell or ability has resolved, you receive priority:

117.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.


117.3c If a player has priority when they cast a spell, activate an ability, or take a special action, that player receives priority afterward.

That means you can keep activating abilities as long as you want before anything resolves. That allows you to stack your abilities in any order you like. When any of the Aphetto Alchemist's abilities resolves and your Coastal Wizard becomes untapped, you receive priority and can activate Coastal Wizard again, before any of its previous activations resolves. At the end of it all, you have 3 Coastal Wizard activations on the stack.
For the second part of your question: Since returning Coastal Wizard is an effect of its ability, not a cost, you only return it on resolution. Since returning the Wizard and the targeted creature are independent effects (no intervening if-clause), it doesn't matter if the Wizard is no longer on the battlefield by the time the ability tries to return it. The targeted creature will be returned anyway.

609.3. If an effect attempts to do something impossible, it does only as much as possible.

Note though that if all targets of your various Coastal Wizard abilities all became illegal by the time they resolve, for example through Veil of Summer, the abilities would all be removed from the stack because all their targets became illegal, and you wouldn't have to return your Wizard either.