1. I understand the use of hashing in PoW, but what's the point of using SHA-256 to generate bitcoin addresses? Security doesn't look too important here, as we obtain these addresses from public keys, not the private ones.
The standard for sending transactions at the beginning was actually directly to the ECDSA public key (P2PK). It was changed only after awhile to public key hash. The main advantage to using SHA256 as a method to send address is that it is way smaller than public keys and hence more convenient to store and type. The added advantage is that there isn't any known attack against SHA256 as of now but ECDSA is susceptible to quantum computer attacks due to Shor's algorithm.
Since your public key isn't exposed till you spend your coins, your address would likely be safe against it.
2. What are the advantages (or disadvantages) of the newest generation of SHA3 Keccak hashing function vs SHA2-256?
One obvious disadvantage is that it would require a hard fork to implement it in Bitcoin. I can't think of any significant advantage for Keccak over SHA256 since there isn't any glaring vulnerability about it right now.