my ps3 has partial ps2 playback, some games dontwork, would that thing fix it?

That's for installing PS2 games that require HDD like FFXI.

does jak and daxter 1 require that?

No

FFXI is the only PS2 game that requires it.

Doesn't it also give you the ability to play PS2 games online using your PS3??

No the BC sku PS3 takes care of that all on it's own. FFXI uses the Playonline app which runs off the network server for PS2. The system data from PSN is just the config settings data to allow Playonline app and FFXI to run on the PS3.
You download and set up that data, then install the game (which takes up 11 gig if you have all the expansions like i do) and boom you have FFXI on your PS3

 missed your post, and yes i believe it does allow ps2 games to play online as well.  The few games left that is that the servers are still up and running.