For RSA, does $p^e=p$ and $q^e=q$ hold? Is it true that $p^e\equiv p \pmod n$ and $q^e\equiv q \pmod n$ for RSA?

Here $n=p\cdot q$, $p$ and $q$ are different prime numbers as required by the RSA system.

I have to find four examples of $w$: $w^e\equiv w \pmod n$ that always hold. My candidates are $p,q,-p,-q$.

It works on examples but I can't prove it.
Is it true that in RSA $p^e\equiv p \pmod n$ and $q^e\equiv q \pmod n$? 


No, it is not always true (and, for realistic RSA modulii, it is rarely true).  To take an example, if we have $p = 11$, $q = 17$ and $e = 3$, we have $p^e \equiv 22 \not\equiv p$ and $q^e \equiv 51 \not\equiv q$


  (I have to find four examples of $w$: $w^e\equiv w \pmod n$. My candidates are $p,q,-p,-q$.)


Actually, there are always at least nine examples, however those four aren't it.  I'll give you three for free: 0, 1, and $n-1$, all you need is a fourth one...