RE: Antenna question
Welcome to the forum
Yes, you can solder and shrink wrap the antenna and it will function just fine. You'll never get it back through the antenna tube but it will work just fine.
Yes, it should be as close to the same guage/size/flexability as you can get to the original. Not really for it's electrical properties but more for it's physical properties like flexability
You can get replacement antenna wire at most any hobby shop that is precut to the right length.
As long as you keep the antenna length close to the original length you won't suffer any reception loss.