If I just checked my pet into the lodge will he become happier? Soon? Or eventually?

not sure where you mean. The Lakeside Resort and the Rathskellar Inn are different places and work differently. Neither one is called a lodge.

The Resort should do happiness within a few hours, it goes up gradually. The Inn doesn't do happiness.

Yup, I'm not sure how long it takes at first since I don't pay attention but the resort will keep him full and happy until he checks out c: You can also give him a minion to keep his happiness full, if you need to feed him for achievements.

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EDIT Yeah, the Inn keeps pets happy too :l

I put my pets in the Inn because I like the game-mechanic better than the Resort one. I like to do experiments, so I took Hay You away from Niketriumphant not too long after your post, and now it's been hours later and she isn't any unhappier. You are correct! Thank you for this, next time I try to do Major Drills I will remember I need to do it with my eaterpet that's not in a hotel.