Is there a base numbering system in which pi has an exact value? How about e?

This really depends on what you mean. Pi and e have an exact value in any base system, but no finite representation. Pi and e are irrational, so no finite sequence of rational numbers (which have finite representations in integer bases, sometimes repeating) would equal pi or e (pi and e are also both transcendental, so there is no polynomial with integer to coefficients that has pi or e as a root). Therefore, only in certain transcendental bases does pi have a finite representation, most notably base pi (where pi is 1), but also bases a*pi where a is a rational number.

No

sure, pi in base pi has the exact value 10