Indeed it does, but it's pronounced to rhyme with "jail" whereas if this would be a Gaelic word, then the initial "G" would probably be hard, as in the words "Gaelic/Gaeilge/Gàidhlig" themselves. It's just old English.
Since this is a perfective verb, does this mean "visit me once" rather than "visit me from time to time"?
Yes, that's true, but in this context there is room for interpretation.
I don't believe that the speaker actually means "only once and that's it". It's just a safer way to ask whether there is a chance of at least one visit. Depending on the relationship of the two, będziesz mnie odwiedzać? might be a lot to ask for, so it's tactically safer to understate the desired magnitude/frequency of the action at first, but then maybe ask for another visit, once the first one is over.