If TMs aren't, then HMs are most certainly not. However, I will permit the moves they teach to be moved by pokémon that should be able to learn them naturally anyway (i.e: A bird learning fly.)
As for the likes of cut, if the tree is big and your pokémon is slow, this roleplay works more or less on the same rationalizing as the anime: You could just, destroy it. 

Also, I was about to close character sheet applications, but since you've got a Ghastly Bespoke display picture, my heart has swelled three times. Both of you can apply, if you'd like, but it ends with you.