The Allied Infantry would capture the City as per normal, the garrison can't move so it gets wiped out in any attack, they don't retreat. It's then up to you to retake the City.
Having these units to protect your Cities, buy you a little time to bolster your defence, if they are attack by just one enemy unit, they will take damage but survive, but if attack by 2 or 3 Infantry then its curtains I'm afraid.
NB: Since these can't move, you can't swap there position with another friendly unit, your other units have to help drive the attacking forces away, if the City garrison falls, the city will be capture. Hope this answers your question more fully.
Yes, but I could/would have issues not having access to the city with other friendly units. As far as "movement" goes, the Home Guard would be a hinderance.