Next time that you have a bunch of questions to ask at the same time, it wouldn't be a bad idea to put them all in the same topic lol.
Anyways, HM 01 (Cut) is in the Ilex Forest. To find it, you have to chase the Farfetch'd that is running around back to it's trainer. Simply walk up to the Farfetch'd, talk to it, and it will run away from you. When it stops, go around it, you should find a PokéBall with a Revive in it. Now, talk to the Farfetch'd again. This should send it towards the entrance. Now, just keep chasing the Farfetch'd until you make it run into it's owner. Once you get the Farfetch'd back to the owner, he will give you HM 01 (Cut).
Also, go back to Azalea Town, and go to the Charcoal house. It is the house south of the PokéMart. The Farfetch'd owner will be in there, and if you talk to him, he will give you Charcoal. When Charcoal is held by a Pokémon, all of it's Fire type moves will do 10% more damage.
EDIT: LOL fixed the mistake. Thanks X1337!!