If I compute the SHA-1 of all possible 40 digits hexadecimal strings, will I get all possible sha1 hashes? Wikipedia says sha1 produces a 160-bit (20-byte) hash value. So if I compute the SHA-1 of all possible SHA-1 values, which as far as I understand is at current time impossible both to calculate and store, will I get all possible SHA-1 values? Would I be able to collide anything?
No. And it's probably a good thing that that's not the case.

All cryptographic hashes (inc. SHA-1) are designed to have no obvious correlation between their input and output. If there is too much of a correlation, then they are considered broken.

If each string of 160 bits produced a different output, that would be a correlation.

That would also mean that all cryptographically secure pseudorandom number generators would have a small bias.

[It's very possible that there are some outputs that cannot be produced by SHA-1].