Welcome to the forum! You will be happy with that camper.
I have a 2008 Surveyor 291. Here is what mine uses....
The tire size will be on the data plate (front street-side corner of the trailer), but mine are 205/75 R14 rated for 1,760 lbs.
As far as battery, I suspect it is 12V unless the previous owners modified it for golf cart batteries, so the answer to that question is, "Any 12V deep-cycle wet-cell (or Marine wet cell) battery that will fit in the battery compartment." Don't buy an AGM or gel battery. The charger will not work safely with one of those types of batteries. That's not a problem though, since the wet-cell batteries are the cheaper ones. My camper requires a Group 24, but I am hoping to upgrade it to accept the much larger Group 31 before the end of the summer.
2008 Surveyor SV-291
2003 Ford 4x4 Expedition
Reese SC WD System 800/10,000 lbs
1976 Cessna 177RG
Wife, son, twin daughters, and mother