Customers who always look for the cheapest are a nightmare imo, sell your service and don't be afraid to charge a fair price for it. I would have more respect for someone charging me a good price for a good job.
Hi, Yes if you connected something external to a zone it would activate on set.
I personally would look to install a stand alone system, something from here maybe.
Duty bound, yes if they pay or arrange it on say the last maint visit. As above you are protecting yourself against a claim.
The customer owns the system but the programming could be deemed intellectual rights of the incumbent.
I would want to know about any tampering of a system I had under contract. Sounds like you are on a winner if they are calling you out to crash and reset imo, but I wouldn't see them as long term customers.