Jimmie, In the interest of getting you some answers, I can tell you how we do it.
Can you back your database up locally if you want to?
Absolutely. We encourage the Doctors to pull down a backup regularly in the event of firestorm, hurricane, etc so that they have a local copy of the database when power and Internet are disrupted.
If your nurse is connected at the same time as you are, does that kick you off with one user?
Depends on how it is configured. We have some practices that only want each long used once, and others where they want to just use a pool of connections. In that case you'll get notified that there are no more connections left.
Also, if one uses lytec for scheduling and billing on site, do you see any issue with using AC in the cloud integrating with the on site lytec?
We have clients with all kinds of interfaces or connections, some run across a VPN to another location, some run from the server instance - it just depends on what the practice needs.