All vehicle batteries (car batteries, truck batteries, small storage batteries, storage batteries, motorbike batteries) will charge with they're mounted to a vehicle that a solar panel is mounted on, provided the weather is Clear or Sunny and it's daytime.
Vehicle welding rigs require fabrication and electronics at skill level 4, and a recipe that comes in most of the low-to-mid range crafting books for those skills. (libraries, mansions and possibly schools are the best places to start looking). You can technically mount them on a vehicle with solar panels and batteries, but the weight increases fast and since you can only have one solar panel per tile (and panels are kinda crappy at charging batteries individually), it's not very feasible to have a welding rig cart with it's own solar panels attached. That was my recommendation
here to charge the battery on an immobile platform while you cut the rest of the vehicles apart, so you can have a charged battery to stick on a shopping cart frame with the welding rig to power it once you have the rest of the vehicle parts ready to re-assemble. You could also leave the panels and charging battery on the car you're hacking apart, but then you couldn't do anything with those pieces to get them out of the way and rebuild/train up mechanics that way.
If you stick wheels on it, will it be easier to move?
Yes. Wheels lower the speed penalty per weight of a dragged item, you can check it by loading up a shopping cart and watching your drag speed drop as the loot piles up... then cut off the casters and try the same.