One thing no one has mentioned: You need to support that arm from the INSIDE when working on straightening it, just as you would when bending it. That means removing it, plugging one end and filling it with fine sand. Then plug the other end. The sand inside will help prevent it bulging out or caving in "here" when you are pressing "there".
Frankly, this kind of thing is tricky to do; you might want to take it to an expert metalworker, perhaps a jewelry smith or experienced metal model builder. Otherwise, take it very slow and easy, and don't force or rush things too much. Think thrice, before acting once.
As for the tonearm not having free horizontal movement, are you sure it isn't supposed to be that way? A few of these fancy arms have servo-control mechanisms that make them feel a bit stiffer than a simple free-bearing tonearm.
The easy solution is to replace the tonearm shaft, but if you do and the characteristics of the replacement are not exactly like those of the original, it may throw the automatic and servo controls all out of whack. Hopefully, this arm is a simpler automatic that won't be so finicky/complicated.
One last thought: is that arm aluminum, or steel? If it is aluminum, DON'T use heat to soften it. Aluminum just gets hotter up to a point, when it suddenly melts. In between, it doesn't get much softer. Steel, however, can be heated a bit to soften it, before bending.
