Maximal chains have the same length? 1) If one has a finitely generated algebra over a field, is it true that any two maximal chains of primes have equal length?
2) If not, are there any other conditions that allows one to conclude that the maximal chains of primes have equal length?

No, it is not true: a counterexample is $k[x,y,z]/(xz,yz)$.
It is true however for a finitely-generated algebra $A$ (over a field $k$) without zero-divisors .
Indeed this follows from [Matsumura, Commutative Rings, Ch.5, (14.H)].  
There he proves that $A$ is "catenary" (even "universally catenary", a stronger property) .
He also proves a formula which implies  that all maximal ideals of $A$ have height $dim(A)$, and together these results show  that all maximal chains of prime ideals have the same length, namely $dim(A)$.
[He defines "catenary" page 84] 
