Does using Beast Ball count as "taking a prize" for the purpose of calculating damage for Stakataka GX's "Assembly GX" attack?
No, it does not. You are switching cards, not taking a prize.
Source: TPCi Rules Team (2018-08-16)
Say I have taken 5 prize cards during the game, but then my opponent uses Naganadel GX's "Stinger GX" attack to bring both players' prize count back to three prizes remaining. So if I then use Stakataka GX's "Assembly GX" attack on my next turn, how many prizes would I count when calculating the amount of damage to be done?
Even though the Stinger GX attack added cards back into your prizes, you have still taken five prizes during the game. So Assembly GX would do 50 damage plus 250 (five times 50) for the prize cards you have already taken.
Source: TPCi Rules Team (2018-08-09)