We had the same trailer.
Yes the battery charges when on house power (unless there is a problem with the converter).
I generally kept mine plugged in during the summer because it was parked near an outlet anyway. I removed the battery in the winter and brought it inside where I used a battery tender to maintain charge.
Yes, the fridge should work, but mine never did. Would be interested to hear if yours will work.
For some reason, my fridge was wired through the cutoff switch (located at the rear curbside edge of the camper top). When the top is down, this switch cuts the mains power to most of the trailer devices to prevent operation/overheating. BUT everyone kept telling me that the trailer fridge shouldn't have been wired this way. Possibly could have had it fixed, but could never be bothered.
If yours is wired like mine so that the fridge won't run on electric, you could run the fridge on propane. Personally, I wouldn't like leaving it unattended like that all night while closed up. You could also pop the top about a half inch to disengage the cutoff switch.
2011 Jayco X19H (purchased 2015)
2008 Jayco 1007 PUP (purchased new, traded for the X19)
2012 Chev Traverse LT with TP