Is it ok to plug into shore power while the inverter is on.
It is OK. When you plug into the landline (or start the generator) the automatic transfer switch (ATS) switches the inverter powered circuits to the land line (or generator). The charger side of the inverter/charger connects to land line (or generator) power and starts charging the batteries.
The ATS can be set to be shoreline or gen priority.
It is ok. I agree with Roger as above. I would only add, when you plug into post and connect to rv, make sure post breaker is off. After all is secure, post and rv connection, then turn breaker on.
With generator usage, there is a delay at start, and all is hard wired, so yes, no problem there.
See next post by wolf10, the technical truth about possibly extending the life of the contractors.
As infrequently as I plug in or start the generator, in the future, I'll turn the inverter off and charger off. It is just not that hard to do. That said, I've never started generator or plugged in with a/c on.
All correct, BUT, it is far easier on the contactors in the ATS if power source is not switched under high amp loads.