Is it possible for a corp to run an out of faction agenda?

There is no influence cost so I'm thinking "no" but that makes the core set very restrictive in terms of both agenda choice and deck size.

There are generic agendas, but no, the agenda's belonging to the other corps (at least the ones in the starter set) cannot be taken, as they have no influence cost/bar.

Last thing on p24:

Note: Some cards do not have any influence value (this is
different than a card that has an influence value of zero). These
cards are identified by their lack of an influence box. A card
without an influence value cannot be used with an identity card
that has a different faction affiliation.

The expansions will offer a much wider variety of generic agenda and in faction agenda cards. A large number of players are looking forward to being able to finally take Private Security Force out of their non-tag'n'bag corporation decks.