I tried to play with my pet with a toy for the Major Drills, but I noticed it said it would only show the pets that want to play right now.

What factors go into whether a pet wants to play or not? And are there factors on whether or not they will read/eat/ do other things as well?
(I know they only eat ten items a day or so..)

A pet won't play if it's at 10/10 happiness, which can happen from being played with a lot, having a minion, or being in a resort. Likewise, it won't eat if it's 10/10 hunger- it needs time to get hungry again after being fed or removed from the resort so it's not being fed every hour. Both of these decay over time, something like one point per hour I think is the normal rate.

A pet will always read a book it has never read before. Books cannot be read to a pet more than once.