The odds are no. The average stick of gum contains about 5 calories, (one can of coke has about 200) so unless you are eating 20 sticks a day, you end up burning those calories chewing gum long enough.
Not likely. You would really have to chew a lot of gum to get a lot of calories into your system to gain weight, but the added sugar is really harsh on your teeth. I guess you really could gain weight but it would be difficult.
No. Don't see any correlation. How did you think of that relationship?
No. It should not do either of these things.