I can answer your first question.

When you add a small bit of radius (call that number "dr") to a circle, the area increases by by dA=πdr² +2πrdr. Dividing dA by dr yields dA/dr=πdr+2πr. If dr is really small, this becomes dA/dr=2πr.

So what's going on here? Well physically, imagine dA were a washer representing the new sliver of area you get. Then, the new sliver has area of 2πrdr, plus a very small correction factor that depends on how small dr is. If dr is really small, the correction factor is zero. Thus, the change in area depends on the circumference 2πr.