I guess the best way to do this is list them all: 
 
1) I heard in Black you can buy stones, and the only way to get them in white is by people giving them to you in the game. That doesn't seem very fair though. Is there another way? 
2) Is there a way to trade/battle online with random people instead of friendcode stuff? 
3) Is there a finite number of stones in white? 
4) How do I get rid of HMs? 
 
I hope to hear your answers soon, and feel free to post your own questions. Thanks!

1)You can't buy stones in either version, but you can get all of them in dustclouds in various caves and routes. 
2)Yes, talk to the woman at the top right of any Pokemon Center. 
3)No 
4)Talk to the Move Deleter. He's in Mistralton City in the building to the right of the Pokemon Center.