If we are playing the Core campaign, can players pick characters from other Expansions, such as Jabba's Realm? Or are they restricted to the characters that came in the Core box?

Just for clarification-

Heroes (the upgradable Rebel player characters) can be played with any expansion.

Allies and villains (the unique and non-unique figures and groups that can be earned, such as Rebel Troopers, Han Solo, and Boba Fett) can generally be used with any campaign. There are a few timeline restrictions, but FFG may have dropped those for future releases, and there's technically nothing stopping you from ignoring those restrictions anyway. Also, there is a handful of other errata, such as Jabba not being able to be earned as an Imperial villain in Jabba's Realm.

Sorry. Yes, I meant Heroes. Thanks.

No worries!

Yeah, any hero can be used in any campaign. In full campaigns, you include the used heroes special side missions as well- but these are written to be usable in any timeline, really.

The Jabba's Realm heroes are merc themed, to the point that their skirmish cards are actually used by the mercs. But even then, the campaign is coherent with any group.