Yes, some inverter/chargers DO have an equalization mode-- it will likely be manually activated.
It is for a short time ONLY, as it substantially raises voltage.
Absolutely, if you can disconnect house circuits while equalizing, it is best for your 12 VDC PC boards, lights, etc.
The best way to determine that you do need to equalize is to check your wet cell batteries with a battery hydrometer. If readings are not very close to the same (cell to cell) it is time to equalize. Equalization is "finished" when all cells read very close to the same. If this is not achieved within a couple of hours, you have a bad call/battery. Amp rate during equalization is quite low-- under 5 amps.
So, yes, only start equalization on a fully charged battery bank.