Problem 1:Moving with charge

A proton moves in the same direction as the current in an infinite wire. For constant current the Lorentz force on the charge is toward the wire. Now go to a reference frame which moves with the proton. In this frame the proton is not moving. Therefore the proton experiences no force toward the wire.

Not strictly true. If you had a conducting wire and a charge, we could easily use the method of images to find the voltage and take that result and find a force on the proton. There is an attractive electric force here. I hate to introduce special relativity to the discussion, but magnetism is a simplification as it drops out of electric force with relativistic treatment of it's field. The magnetic force that we calculate in the "stationary" frame is just an easy way to calculate the relativistic electric force.