EDIT: To clarify, I am answering part 2.
When dealing with derivatives, the most important thing is to be careful. There are dropped and confused signs, arithmatic errors, and your substitution for v is wrong. (Also, not that it matters, but you kind of pulled your last step out of nowhere: continuing from the step above would give 2x). I'm sorry to say but I have to scrap everything except the mini-derivatives; those were correct.
u'v + v'u =