this is gonna sound like a cop-out, but both depend on the totality of the circumstances.
1. For lying about you, it depends on what the lie is and how it spread. Sometimes it's best to deal with the person head on and confront them with the truth. Sometimes it's better to show someone else what the truth is and let them be your missionary. sometimes just dropping partial hints to others that it is a lie works. Sometimes just ignoring it and letting people see that is is not true is best. And sometimes it's best if you let people believe the lie, cause they feel really dumb and embarrassed when (if) they realize on their own that they were wrong. Guess it kinda also comes down to the nature of the lie, how its spread, and whether you want to bother or not.
2. There is no one answer here either. Sometimes being 100% open and over telling about everything is the right thing, try to show them you have nothing to hide. Some people would come around, but others would only fixate on the fact that you must hiding something as you're telling them "everything else". In some cases maybe not being as open is better, that way they have less to get worried about. I have no idea what the "correct" answer is here, because I really think it just depends on the person who is jealous. They either get over it or they dont.