To the first part:
Ethereum uses the deterministic (and elliptic-curve) variant of DSA. So, unless you don't switch manually to the probabilistic variant, you'll get always the same signature.
To the second part:
This is a bad idea because:
- if you use the probabilistic variant of (EC)DSA you'll need to feed it with some randomness that should be better used directly instead of extracting the randomness from the signature and reusing it afterwards to derive the keys.
- reusing the randomness of a signature makes the signature itself useless because the keys that you've derived by using the signature's randomness will become easily derivable just when you publish that signature to someone else.
Update:
There is a deterministic variant of DSA and ECDSA that is described under RFC6979. This means that DSA (and ECDSA) can implemented in both ways, deterministic and probabilistic. It was shown by Koblitz and Menezes that deterministic DSA has similar security guarantees as its probabilistic counterparts.
The big advantage of that deterministic construction is that it gets along without the requirement of a good random number generator (RNG) and achieves therefore a higher "real-world security".