What happens if I have a minion that occupies more than one site (e.g. Mountain Giant) and also carries Land Deed?
Land Deed only grants you control of the site that Land Deed occupies.
If I gain control of a site in the middle of my turn, does it produce mana immediately?
No. Sites produce mana at the start of your turn and when they enter the realm. Since the site with Land Deed was already in the realm, it doesn't produce mana merely because took control of it.
You control the Rubble and your avatar will loselife if the Rubble takes damage, but the Rubble still doesn't provide mana. i.e. you own a hunk of worthless rock!