Can I use these all interchangeable?
My tooth hurts real bad. / My tooth hurts so bad. / My tooth hurts really bad.
My tooth aches real bad. / My tooth aches so bad. / My tooth aches really bad.

Question

Answer

Yes, they all are interchangeable ,but gramatically correct: "My tooth hurts badly".   bad + ly ending on adjective ( bad)  becomes an adverb to describe how severe (bad ) the pain is.  Adverbs many times will end in "ly".