I recently moved to a new home and I was wondering if the friends that I had allowed into my old home are still welcome in my new home or do I need to reinvite them?
Your friends were uninvited as soon as you logged off after the first time they were allowed in. They must be allowed in each session that you're both online.
Now if they were allowed into your house during the playing session you moved, I'm not sure if they'd be allowed in the new place after you sold without new permissions.
Someday they shall have some sort of key or something that will allow people into your house permanently (or until you take away that privilege). Today is not that day, though (or maybe it is and the devs will surprise us!). There will also be group halls/streets eventually.