I'll assume that it's got two axles....usually something 10' would have them.
What brand is the trailer...hard to tell much about it from the pictures.
I doubt that you'll have any problems with the golf cart...as mentioned, they aren't all that heavy.
But....not knowing just how that tailgate is, you may want to carry some 2x8 planks to lay over the ramp, so that you drive up something a bit more substantial. My gate is steel, and fairly heavy, and smaller than that one, so I'm questioning it a bit too....but you can get those ramps that you would use for a lawntractor going into a pick-up (from Harbor Freight or HD)and avoid any weight on the back door/ramp.
If the floor is questionable, it's best to replace it. If it's bowed like you mentioned, it may be that the previous owner just laid down a sheet of plywood, and you can pick it up and check under it.
Replacing the floor isn't a huge job, but if you aren't handy, you may want to find someone to do it for you.
I replace the floor in my utility trailer last summer....while I was at it, I cleaned, wire brushed and painted the frame under the wood.
Biggest hassle with the golf card, may be getting in/out of the golf cart when it's in the trailer, so you may want to park to one side.
Good luck