What is the difficulty of DLP in GF(P^Q) with a subgroup with a prime order of L Given a finite field GF(P^Q), having a subgroup with a prime order of L (P,Q,L are all primes), how difficult is it to find the discrete log, is it related to P and Q or is it related to L, or to both. What is the time required? O(sqrt(L)) ?

Does index calculus apply to Galois fields?
First there are the "generic" discrete logarithm algorithms like Shanks's "baby step, giant step" and Pollard's $\rho$, which run in $O(\sqrt{L})$ and are thus of exponential complexity (in the size of $L$). Those algorithms work in virtually any group.

In the special case of the multiplicative groups of finite fields, we have subexponential algorithms (i.e, algorithms which are "in between" polynomial and exponential). The index calculus algorithm has many variants whose complexity varies slightly depending on the relative sizes of $p$ and $q$ (see the Wikipedia page and its references), but the details are not very important unless you want to specialise in discrete log algorithms (in which case, read the references mentioned above). What's important for cryptography is that due to such algorithms, cryptosystems based on the discrete log in finite fields must use larger keys than those based on the discrete log in elliptic curves for the same level of security.

Also, the "quasi-polynomial" algorithm of Gaudry et al. works for small $p$ (basically, when $p \le q$), and such cases should therefore be avoided, but they were rarely used in the first place anyway.