Can I use Salsa20 as a good non-cryptographic PRNG with different streams if I reduce the number of rounds to 8 and omit the addition step at the end? I want to omit the final step because I don't want to get all zero outputs.
Reducing the rounds to 8 would give you Salsa20/8, which is not just a fast PRNG operating at 1.88 cycles per byte on Core2Duo, but is still quite cryptographically secure with the best attack requiring approximately 2244 operations. Removing the final addition step would not be good though, as without that, it would be trivial to reverse the function and discover the key and counter given just a single block of known plaintext. You will not get all zero outputs by keeping the addition, so you should keep it.
You could cut the algorithm down to four rounds in order to roughly double the speed while completely sacrificing cryptographic security. Less than four rounds results in incomplete diffusion, leading to biased and non-uniform output. However, it will still be roughly twice as slow as the fastest dedicated non-cryptographic PRNG, XorShift128+ (an LFSR-based PRNG at 0.48 cycles per byte on Kaby Lake).