Is F' a pseudorandom function when F is composed with G, a pseudorandom generator? If $F$ is a pseudorandom function, is $F'$ also a pseudorandom function in the following: 
$$
F'_k(x)=F_k(G(x)) \space \space \text,
$$
where $G$ is $a$ pseudorandom generator?

Also, does the other directions holds (i.e $F'_k(x)=G(F_k(x))$)?
Certainly yes.
Proof
A pseudorandom function must produce pseudorandom output for any given input. So, any seed value will provide a pseudorandom output. However, this might not work if the function has a limited input domain (list of possible inputs) causing incompatibility with the two functions. In your case, $F$ should be able to process the input provided by $G$, and vice-versa for the second case.
Proof of concept
Let $F$ be a hash function, say SHA-3, and $G$ be AES-CTR, Then:
Case 1, $F'_k(x)=F_k(G(x))$: AES-CTR would generate a pseudorandom sequence which would be hashed by SHA-3, and, needless to mention, cryptographic hash functions are supposed to produce strong pseudorandom output, regardless of what the input is.
Case 2, $F'_k(x)=G(F_k(x))$: The hash digest of $x$ produced by SHA-3 will be used as a seed for AES-CTR, and, once again, block ciphers are also meant to produce strong pseudorandom output regardless of the input.
Verdict
Both the implementations suggested in the Proof of Concept section explain how $F'$ acts as a pseudorandom function. It is important to know that the security and quality of pseudorandomness is highly dependent on the outer function, $F$ in case 1, and $G$ in case 2, as it's output is what determines the required. The underlying function will only work as an internal state generator, where the state could have been anything, not necessarily pseudorandom.
So, building a PRNG by chaining two different PRNGs (as you did in your post) will surely work out.