Is PS3 / PS4 Online Multiplayer Cross-Platform?

It is possible for PS3/4 games to be Cross-Play but it is not the default. You will have to check whether each specific game supports PS3/4 cross play as it's up to the developer to implement.

Not at the moment. Maybe the CoD developers can add that feature after patching it, if enough users request it on CoD community.

Playstation.blog is your best source, considering it comes directly form playstation

Over here it states that the ability for ps3 and ps4 users to crossplay is available, but it depends on the developers.

Q: Can PS4 and PS3 users play together online, cross-platform?

A: The functionality is technically possible and some developers may choose to support it for their titles.