Gingivitis is an infection of the gumlines where the teeth and the gums come together. The bacteria that are primarily involved in causing this infection are specially adapted to this environment where they are protect from the high levels of oxygen in the air. Therefore, these bacteria don't do well with air exposure and they would be not likely to cause an infection elsewhere, such as on the lips.
I would suggest that you go see your primary care doctor
to help you figure out what is going on with the chapping of your lips. With the vigorous brushing, etc that you are doing for your mouth it may be that you are irritating your lips (for example, with mouthwash or other chemicals). If so, then this should respond simply to using a good chapstick frequently and avoiding licking your lips or picking at the dry skin.
On the other hand, if the dry skin and cracking is more localized to the corners of the mouth, then this is a condition called angular cheilitis, which often requires antifungal medications in order to clear up entirely.
Your primary care doctor will be able to help you figure this out and will prescribe any necessary medicines.