Thanks for the photos. Your kart is in really good condition. Seat looks like a bargain but only if it fits.
You know, you can use that non-matching wheel on the front left until it breaks (if it breaks)---it's just that everyone freaks out on the "Harbor Freight" wheels or anything that looks like them. There were multiple people on this forum long ago and a guy on Youtube in particular that used those wheels on his kart and they lasted 10 minutes or less. The greasy vinyl "rubber" and the thin stamped metal construction weren't made for stress. Your wheel looks a bit more substantial or designed to take some abuse.
No worries on the storage issue, it is what it is, but a tarp or cover is a very good idea. You could also remove the roll cage/brush guard (it is only held on by screws) and then your tarp can be smaller. Watch out for rodents making a nest inside your engine, like around the blower shroud/recoil starter and around the cooling fins. That's bad news for the engine because it can overheat.
There's debates on the merits and detriments of roll cages/roll bars on a go kart and if they really protect the riders. This forum says if you have a roll cage, then you need to use the seat belts but the belts on your kart, for example, are just a combination lap and shoulder belt (automotive style) and they won't really hold you in in a roll over. Just be careful out there and be aware of the dangers.