There should be some offset to the wheels, if the "axle" is long enough add maybe an inch of spacer, could use washers, and make sure you can get the nut back on. Just because the rim is 5" wide doesnt mean a 4" long axle wont work. If worst case forces you to make the axles longer they are just bolts welded in place and your bearings do all the work, cut the old ones off where the head of the bolt and the frame meet, and weld longer ones in place, make sure they are straight and you have some nylon lock nuts that fit them. Longer bolts will allow more spacer room to get the wheel away from the frame to fit.
A grinder with a cutting blade will do fine...
Any welding machine will do.. MIG or fluxcore wire feed will will be just fine...
So with the flux core, theres no need for additional gas right? I wonder if I could get it done with one from HF
You certainly could. Or just as easily take it to a local welding shop and save time and money and get it done professionally. It's all up to you.
There is a learning curve to welding. But it's a good skill to have. My only personal issue with the cheap harbor freight machines is that as soon as you learn how to use the dang thing you'll regret that you didn't spend a little more to get a more capable machine. Other words you will outgrow the machine real fast if you actually use it and enjoy what you are doing.
Still, check into using some spacers first as mentioned above. And whatever you decide, best of luck.