Does SHA-1024 hash exist? Does SHA-1024 cryptographic hash function exist similarly to SHA-512? If not, what's the reason for that?

Links:


SHA-2 at Wikipedia
SHA-3 at Wikipedia
No, because even SHA-512 was considered overkill from a security perspective. It has 256-bit collision resistance, which is unbreakable. (The link is about keys but a similar argument applies.)

If you think large quantum computers will be efficient, a 512-bit hash makes some sense, but even then a 1024-bit one wouldn't. A quantum computer requires $O(2^{n/3})$ operations to find break the collision resistance of a secure $n$-bit hash.

Finally, SHA-256 uses 32-bit words and SHA-512 64-bit. If you wanted to use the same way to extend SHA to 1024-bit state, you would need to use 128-bit operations which are slow on practically all platforms.