LBJ
(LBJ)
1
How do I transform selected vertices into a perfect circle?
Install looptools add on (included with blender)
In edit mode select verts and select circle.
Hope that helps.
LBJ
(LBJ)
3
How do I gradually transform each segment of this pipe elbow into a perfect circle?
The last edge loop has to be a perfect circle, but the the other 3 need to gradually transform into a perfect circle?
How can I do this without creating any ngons?

Have you tried looptools?
There is an influence slider for the amount of circleness you want.

Best is to keep only the first and last segment (last being a perfect circle) and do a bridge between them.
either way, looptools is the way forward.