Yes.
No, it would charge the batteries and then they would discharge through the inverter to supply the house. Or go straight to the inverter if the batteries were full.
Depends on the charge controller, could be interfaced to the Sunsynk, could be a third-party integration e.g. via Modbus or Home Automation, could be a separate readout from the charge controller either local or over the web..
Simplest control stragegy is the charge controller seeks to achieve/maintain a defined float voltage on the battery (?3.45V/cell or something, my Pylontecs are 3.47 to 3.50V when fully charged), subject to a charge current limit you can define. This adds to the current the Sunsynk can draw to feed the house and/or prolong its duration. No changes to what the house sees apart from that.
The Sunsynk might be upset by seeing changes in State of Charge it did not know about/cannot rationalise, or the reported values may not be correct. If there is a Sunsynk user group on the web someone on there will know if you can do this. Not my field really, in Victron world it is all done for you, you get what you pay for(!). Have customised it a bit by writing flows in NodeRed. Family also gave me an Arduino kit but I have yet to get to grips with it properly. HP install has kept me busy this year.