SHA-1:Is there any mathematical result that gives us the minimum number of 1's in a 160-bit SHA-1 hash output? Is there any mathematical result that gives us the minimum number of 1's in a 160-bit SHA-1 hash output? What is the probability that a 160-bit SHA-1 hash output contains at least 128 1's?
No, theoretically a SHA1 hash can be any 160-bit value, including the string of 160 zeroes. 

As for your second question, if we fudge a little bit and consider SHA1 a truly random function this becomes the same question as the following:

If we flip 160 coins, what is the probability that at least 128 of them will be heads?

Solution is left as an exercise to the reader; I suspect this might be a homework question.