No you can not
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No Pokémon are not capable of evolving while the Daycare Couple are taking care of your Pokémon you will need to take them out of the Daycare in order to evolve your Pokémon.
No
You have to trade with a friend locally by going to the top floor a Pokemon center and going to the trade center.
Your Pokemon cannot evolve while in the Day Care.
no, they dont.