Pseudorandom function or not? I've read Intro to Modern Cryptography PRF section (and example 3.26) and on stackexchange(1, 2), but I still don't fully know how to prove it (with probabilities)

To prove it's a PRF, I know you have to prove that no adversary can distinguish $F'_k(x)$ from a random function or I show a possible attack. So the question is (similar to 3.10 from book):


  Let F be a length-preserving pseudorandom function. Prove $F'_k(x): \{0, 1\}^{n} → \{0,1\}^{2n}$ such that $F'_k(x) = F_k(0^{n})||F_k(x) $ is a $F'_k(x)$ keyed pseudorandom function or not. Where || denotes concatenation.


What I've done so far: No. Consider an adversary A that input $r \in \{0, 1\}^{n}$, it returns 1 if its first n bits are 0...0. On a truly random string it returns 1 with prob $2^{-n}$. On a pseduorandom string it returns 1 with probability 1. Then adversary has advantage $1 - 2^{-n}$ which violates the definition of psuedorandomness (as it's no longer &lt; negligible?). 
1. I believe this is correct, or what have I not understood correctly?
2. If it is the case, I am having trouble applying this to $F'_k(x) = F_k(x)||F_k(\bar x) $ on a paper I found here, example 3.7.2. It says No, but I'm having trouble connecting it with the method I learnt in class, but it looks random to me. I would say it returns 1 if its last bits?? I'm not entirely sure.
There are actually two different ways to do this, I will give hints for both, with additional hints in the spoilers:

First approach: Think about the definition of the security game again, and how many queries can a distinguisher make to the oracle?


   You can query the function on two values. If you compare those two results, are there similarities? And how would a truly random function behave?


Second approach: As others have stated in the comments, what happens if you query $F_k'(0^n)$?


   Since $F_k(0^n)$ is a PRF, you can just assume that this one is actually a truly random function. What happens if you query a truly random function twice? Is that visible in some way in $F_k'(0^n)$?


From your last comment, I guess this isn't clear yet: The distinguisher can query the function for any kind of $x$, as often as he wants - with the limitation of being a polynomial time algorithm. And he has to find out whether the results he gets back are from a truly random function or from $F_k'(.)$, with just $k$ being drawn randomly. So the distinguisher can only do one thing: Choose $x$ in a clever way, so that some structure appears within one result or over several results, if it is $F_k'$ - and the random function does not have that.