Long passwords and Bcrypt
A password manager that produces 16-character passwords is sufficient for most cases. Users who go for 100-byte passwords are usually overly-paranoid, since the actual security benefit is outweighed by the inconvenience. Therefore, limiting a password to 72 characters, while in theory reducing the number of possible passwords, is still very reasonable.  

That said, hashing a hash is a bad idea if the hashes are related. sha512(sha512(password)) is more prone to collisions than sha512(password). However, bcrypt (being built with blowfish) is different enough from sha512 that this doesn't apply (not to mention bcrypt is salted). While you're still reducing a user's password from 72+ bytes to 64 bytes, the expected entropy of a user password is orders of magnitude less than 512 bits. It is safe to assume that sha512(password), while technically shorter than password, has much entropy as the original password.  

Your second idea seems to be reusing a part of the password, so I don't see much of an increase in security. The entropy from p[0:8] will be spread out with sha512 sufficiently enough already.