페이스북을 봐요? => Sounds unnatural
페이스북을 해요? => Fine, but 페이스북/페북 해요? without 을 is more common, especially in conversation or messaging.
핸드폰을 해요? => Unnatural.
핸드폰을 써요? => Okay, but 핸드폰 있어요? (Do you have a cellphone?) is better when asking whether someone owns and uses one.
(Also, people seem to assume everyone has it, so this phrase is probably not used much)
핸드폰 자주 하지 마! => Okay, but it's probably not the best phrase. I'd prefer the following.
- 핸드폰 너무 많이 하지[쓰지, 사용하지] 마 = Don't use the cellphone too much (for the fee, health, or other reasons)
- 전화 자주 하지 마 = Don't call (me) too often.