Is it computationally feasible to find two transaction IDs with the same first 20 bytes? Transaction hashes uses sha256, so they are 32 bytes long. If I have to store a very large database of transaction hashes, only for the purpose of checking existence later on, is it really necessary to store all 32 bytes? If i store only 20 last bytes and I compare transactions using only those last 20 bytes, would the crypto guys get mad?
Is it computationally feasible to find two transaction IDs with the same first 20 bytes?


20 bytes is still 160 bits of security, which is considered very secure. If you are sure that you only need the IDs to check for existence, 20 bytes of the hashes should be fine. 

Hard drive space is pretty cheap, though, and you might need the full TXID someday just for interoperability between other systems. So make sure 20 bytes really does satisfy your use cases. A 20 byte hash is sufficiently secure, though.