Does RSA - the company - know the $p$ and $q$ used to create $n$ ($p$ and $q$ being prime) With the RSA cryptosystem, you  publish $n=pq$. Does RSA $-$ the company $-$ have/store the values of $p$ and $q$?

If not, how do they find $n$ without knowing $p$ and $q$.
Also, how do they find $(p-1)(q-1)$ without knowing $p$ and $q$?

In short, should I trust a private company with my encryption?
In short , should I trust a private company with my encryption?


In short, no, you shouldn't and you don't have to. That is the beauty of the RSA encryption algorithm (which is completely separate from RSA the company). You can use RSA the algorithm without RSA the company knowing anything about it.

If you want to communicate securely with RSA the company using RSA the algorithm, you tell RSA the company your RSA (the algorithm) public key, which includes $n$ (as well as the encryption exponent $e$). They don't need to know $p$ or $q$ or $p-1$ or $q-1$.