How unique and secure are verification codes created from different hashes?
There should not be any need to use several different hashes. One such as SHA3 will provide what you need. A Hash, Salt, and Stretch configuration as provided by for example bcrypt or scrypt will add extra protection if you want it.

As for shortening the code, yes this will decrease security but the same length of a NIST recommended password should suffice: 8 characters