If the converter is running properly, just leave it plugged in. It will stop charging once full. If you unplug when you leave, there likely are small loads dragging the battery down during the week and then on the weekend, it needs to do a big charge which can take hours to get all the way to 100%.
At 5yr old, not surprising if the battery needed replacement, so hard to say if there was a problem or it was just old.
Do a little online research to find out if your converter is suitable for a Gel battery. They have a different charging regime and the wrong can quickly damage it. Assuming you are typically plugged into a pedestal, just get a basic flooded lead-acid battery, no point in getting fancy.
Yes, the cooling fan will run when charging the battery or when there are other significant 12v loads. When running 12v devices, it will cause the system to go into charging mode. Something small like the air/con thermostat by itself doesn't draw enough to trigger this, so if the battery is fully charged, the lights are off and no other 12v devices are in operation, you shouldn't hear the fan running (it might kick on once in a while for a short period but shouldn't be running a lot)