In which case number field sieve/index calculus is faster for solving discrete logarithm? Given the normal discrete logarithm problem:
$$a = b^c \mod{P}$$
with prime $P$ and numbers $a,b,c$
For which kind of $P,b$ the NFS/IC algorithm is faster than Baby-Step/Giant-Step+ Pollard's Rho ($\approx \mathcal{O}(\sqrt{q}) $)?
(with $q$ the biggest prime in factorization of $P-1$, with $P$ big prime)
Or in which cases NFS/IC it used?
Using its notation, the question is about the difficulty of the Discrete Logarithm Problem in a Schnorr Group modulo $P$, of prime order $q$. I'll assume $b^q\bmod P=1$ and $b\bmod P\ne1$.
That DLP problem is finding $c$ chosen at random in $[0,q)$ given $P$, $q$, $b$, and $a$ obtained as $b^c\bmod P$. Depending on parameters, the best known algorithms fall into two complexity classes:

somewhere between $\mathcal O(\sqrt{q}\,\ln P\,\ln\ln P)$ [in theory] and $\mathcal O(\sqrt{q}\,\ln^2 P)$ for Baby-Step/Giant-Step and it's practical improvement: Pollard's Rho with distinguished points (which can be efficiently distributed and requires little memory; see Paul C. van Oorschot and Michael J. Wiener, Parallel Collision Search with Cryptanalytic Applications, in Journal of Cryptology, 1999). The cost is often stated as $\mathcal O(\sqrt{q})$ multiplications of integers of size $P$, and this has been recently shown to cost $\mathcal O(\ln P\,\ln\ln P)$, see this.

something like $\exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\right)(\ln P)^{\frac{1}{3}}(\ln \ln P)^{\frac{2}{3}}\right)$, for the Number Field Sieve applied to the Discrete Logarithm (see this).



in which cases is NFS/Index Calculus used?

For a given size of $q$, the first class of algorithms (Pollard's Rho..) is best for large $P$. The second (NFS) is faster for relatively small $P$, including $q$ a Sophie Germain prime (equivalently, $P$ a safe prime).
For 256‑bit $q$, the first class of algorithm is better for 8192-bit $P$, the second for 512‑bit $P$. I prefer not digging where exactly the crossover is, or what's the exact difference between NFS and IC.