To check the balance,the person who will be driving it should climb in WHILE ANOTHER PERSON STANDS BETWEEN THE SHAFTS and holds them right where they would rest in the tug loops.That person should be able to hold up the shafts with little effort, because if balanced, there will be only a pound or two of 'load' on each shaft. It may be even better for the prospective driver to hold the shafts while someone else about their weight climbs into the cart; that will give the driver the direct experience of how much weight there would be on the shafts.
Can't really tell, but it looks as if that 'might' be a PONY sized cart,similar to the original 'Frontier' brand. Pony sized usually had 60" shafts, and more distance between the shafts at their closest point: a pony sized cart would be too large for 32-34" minis. The standard length for minis in that style of cart is 48", and shouldn't vary much from that, in order for the horse to be a correct, safe distance from the front of the cart,and the shaft ends to extend the proper distance, which is just to the point of the horse's shoulder, when harnessed and hooked.
Those should indeed not be bicycle wheels; if in doubt, see if you can take one to a certified bike shop for their opinion, before committing to buy, as you definitely do NOT want to use it with bicycle wheels. They do appear to be 20" wheels, which was the standard size for the mini sized original Frontiers as well as most other brands…but at least sometimes, also the size used on pony carts.
Good luck!
Margo