Are the SHA family hash outputs practically random? Say I hashed the output from a random number generator (with nonce), would the resulting SHA256 hash be as random as the inputted number?

And If I used the first 5 hex characters, and then used the first 5 ignoring the very first character and so on repeating removal of the first character, would each number be just as random?

Example of what I mean, here's a 512 bit hex string.

26657320797d4f5b385d43274a246178263f3b686b645e375b45643442


Is any segment of 5 characters chosen from random locations in the string guaranteed to be a fairly random number?

I am trying to make a pretty simple random number generated based on a hashed seed.
Say I hashed the output from a random number generator (with nonce), would the resulting SHA256 hash be as random as the inputted number?


Let's suppose you flipped a perfectly fair coin. You flip it 1024 times to create a bit string of 1024-bits. 

Because the coin is perfectly fair, this means that each strings of 0s and 1s will appear with precisely the same probability.

What happens if we apply SHA-512 to these strings? Well, in order for SHA-512 to give a completely uniform 512-bit output, it would have to be the case that if we hashed every 1024-bit string then each 512-bit hash value would have to appear precisely $2^{512}$ times. 

There is no proof that SHA-512 behaves in this way. I would be very surprised if this turned out to be the case! 

As such, the output of hashing a truly random stream is a stream which is bias. This bias will be very small indeed, but it will be there. So the answer to that specific question is no.

In practice, the bias is so unbelievably small that it doesn't matter. In fact, many hardware RNGs hash the entropy pool to produce output bytes. 

However, it's worth noting that using such an RNG as the basis for generating a  one-time-pad actually ruins the security proof! It would be technically insecure but good luck exploiting it.