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)
Do I use Togekiss' "Precious Gift" Ability before or after taking prize cards at the end of my turn?
Precious Gift is used at the end of your turn after attacking and checking for a KO but before Pokemon Checkup. So if your attack causes a KO you would take Prize cards first, and then you can use Precious Gift.
Source: Obsidian Flames FAQ Addendum; TPCi Rules Team (2023-08-10)
Do I have to attack in order to use Togekiss' "Precious Gift" Ability?
No, the "(after you attack)" text is just a reminder that your attack happens before the end of your turn. You can still use Precious Gift even if you don’t attack.
Source: Obsidian Flames FAQ Addendum; TPCi Rules Team (2023-08-10)