How did people get control of my account?

First off don't worry too much, no-one is controlling your account. There's a couple of things going on, but you are the only one in control of your account, rest assured

"I was just reported by some obviously lying troll(?) saying I insulted his question with a black cartoon statement . Of course I've appealed but this isn't the first time."

If you did post an answer which was directly insulting the asker then to be honest the chances of getting it reinstated is slim to zero. Direct insults are considered to be violating Community Guidelines so appealing will only result in you losing a further 10 points for a failed appeal. It's best to just suck it up and treat that one as a learning exercise (how not to get reported next time)

"Since we never know if the appeal is successful, how can I keep them out of my account."

You "should" get a reply from Customer Care but unfortunately the reality is that they don't always reply. If you don't get a reply after a week or so then follow up again using the link at the bottom of this page

"I frequently find other names in my response record. How do they do that?"

You are presumably using the Mobile Version of the site and in which case you (and many others) are seeing a glitch where the name of someone else's account appears as yours in the account info data. Whilst it looks like it;'s someone else's account, then it's still yours and it's only a display issue. You cannot change their account data and they can't change yours.

Close account and make new one