transform a selection into vector path

Sure... go in to your "paths" pallet and at the bottom of the pallet there is a "make working path from selection" Icon. it is the forth one from the left. You can then copy and paste into Illustrator

 

Yes.

In Photoshop,

Turn on the Paths palette
Click on the options button and select Make Workpath
Set the tolerance - lower nuber is tighter tolerance and greater number of nodes
Save the path
Repeat as needed
When done, File > Export > Paths to Illustrator

The result is an AI vector file with no fills.

In Illustrator, you simply select Trace with the bitmap image selected. This is normally done globally and then node/path edited to reach a usable result.

Neither method is particularly good when compared to hand tracing with a Bezier pen.

 

You will need to use your path selection tool. hit (A) on the keyboard an it will bring you to that tool