If I use Boost Shake to evolve Togetic into Togekiss, can I use Togekiss' "Precious Gift" ability, or does the effect of Boost Shake prevent me from doing so?
Precious Gift takes place at the end of the turn, so you still get to do it after using Boost Shake.
Source: TPCi Rules Team (2024-01-11)
Can I use two Boost Shake to evolve my Basic Pokemon to a Stage 1 and then evolve it again to a Stage 2?
Since Boost Shake ends your turn immediately after using it, you cannot use two of them in the same turn.
Source: TPCi Rules Team (2021-09-09)
Boost Shake says it can be used during my first turn or on a Pokemon that was put into play this turn. What if my Basic Pokemon was in play last turn but I evolve it into a Stage 1 this turn; can I use Boost Shake on that Pokemon to search for a Stage 2 and put it onto that Pokemon?
Yes, you can.
Source: TPCi Rules Team (2021-09-09)