Can Pseudorandom values be used as secure keys? I want to derive $n$ keys from a master key. 

I pick a master key $mk$ and use pseudo-random function (PRF) to generate $n$ pseudo-random values, $v_i=PRF(mk,i)$ where $1\leq i \leq n$.
Next, I use key derivation function to generate $n$ keys: $k_i=KDF(v_i)$, where $1\leq i \leq n$. Note that key derivation must be deterministic. 

Question 1: Is above scheme a secure way of generating $n$ keys?

Question 2: Can we consider values $v_i$ as secure cryptographic keys?
Yes and yes, as mentioned in the comments.

It is worth noting that Bitcoin wallets use a scheme similar to this in BIP32, a method of creating n various EC keypairs from a single seed deterministically:

https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki

EDIT: This answer only holds true if  is a CSPRNG.  (This application is sort of why the definition for CSPRNG exists.)