1)
来ないか is an expression for invitation. Thus, 大樹の周囲を片付けるから共に来ないか is the content of the invitation letter.
2)
Yes for こよう.
3)
"The -te form of causative + もらう" is a hamble expression for "to do", i.e., 行かせてもらう means "I (will) go" with connoting "it's for my benefit, so you don't need to be concerned about it".
"I'll have you do ____ for me" is for "the -te form + もらう"(行ってもらう).
4)
It can also mean "well then, I'll go".