Welcome to the forum, Kelly!
The best way to check the axle
is to jack it up in the frame just behind the axle
and watch whether the axle arms drop down as the trailer rises. Alternatively, have an adult jump inside and watch for movement of the axle arms. You're trying to find out whether the rubber inside the axle tube is hardened, indicating a worn out axle.
Do a google search on "rubber torsion axle" to better understand how they work.
Floor rot is the biggest thing I'd inspect for. Look carefully inside all the cabinets and benches where there is no finish flooring. Feel for soft spots where you walk and look underneath as well.
As to the rest, you can download a buyer's checklist on this site. It's in the "More" tab at the top right of the page in the "Document Center." It's always a good idea to take a second set of eyes who is not emotional involved in the purchase.