The first one is right. Because, like mr. dzeri said, you have to use the infinitive form, because you already have the verb "don´t want" in a tense. :)

Both sentences are correct, which one you will use depends on the context, but the first one is more common because to+inf is usually used after verb 'want'

They are both correct and there is only one context - you have a piece of information or a situation that you want to keep from the outside world. The second one is a little bit more dramatic than the first and you might hear it in a movie, for example, where a family finds out their teenage daughter is pregnant. The father could say "We're sending her to live with her grandmother until the child is born - I don't want anyone around here knowing (about) this!"

Both right but with different meaning depending on the context you are using them in !

Yes there is a difference between the sentences. The first one is correct i.e "I don't want anyone to know this."