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 its field. The magnetic force that we calculate in the "stationary" frame is just an easy way to calculate the relativistic electric force.

The assumption in this problem that the velocity, v, is frame dependent is not valid. It has to be in reference to the wire. This is very similiar to a friction problem that is also velocity dependent. If an object is sliding on a surface, it cannot be suggested that there would be a different force in the object's reference frame because v=0. The object's velocity must be referred to the surface it is sliding over, or in this case to the wire.