the relationship between gums accesses and root canal accesses can be complicated. they are called "endo-perio abcesses" and come in a wide variety of involvement. More would have to be known to answer that. some do and some don't.

That depends on whether the source of the abscess is coming from an infected tooth.
If the cause of a a gum abscess is from gum disease, periodontal condition, then a root canal may not help.

Yes - it should take away the source of the infection

I'm assuming the "gum abscess" you are referring to is a fistula or gum boil related to the infected tooth. If this is the case, getting rid of the infection with a root canal will result in the fistula disappearing as well. It may take several months for this to disappear after the root canal is complete.

If the gum abscess is coming from the tooth that had the root canal, then yes, it should go away.