Possibility of contructing a valid transaction without knowning private key
Yes, but that is hard. This is known as a collision, and you would be finding a collision in SHA256. Because you would be reusing a known hash, what you would be doing is performing a preimage attack. Since SHA256 has no known preimage attack for all 64 rounds done in a normal SHA256 hash, you would need to do 2^256 hashes in order to be guaranteed to have found a colliding preimage for that hash.

This is why a cryptographically secure hash is used for signing operations.