In order I'd look at physical security (supplimentary locks to doors and windows), then an alarm, CCTV rarely deters it just means you can watch the covered faces whilst they ransack your home.
I'd have thought a caveat disclaimer would be the norm?
Remember all customers are liars and will deny you ever discussing it, or the person will have moved on etc. etc.
Yes, but better done and tested by the engineer, especially as you have paid them to do it imo.
If they are flagging a tamper at the panel (lid not fitted correctly) then they are communicating.
The charging cct struggling to charge a faulty battery can cause all sorts of issue, better replacing and seeing how it goes, probably OK once swapped tbf.