Could an infection in a sinus cavity result in the infection spreading to the upper gum area close to where that sinus cavity is? could that cause the gums to swell as well?

Probably not. The infection would have to break through the wall of the sinus which is highly unlikely. Please see a local dentist for an exam... it appears you have a dental issue.

Your sinus is not connected to your gums. If your gums are swollen it's from oral hygiene that needs to be addressed. See a dentist for a deep cleaning and check your medications if any, some can cause swelling.