If you'd been reading the first couple of issues, you'd not be so ranty about it.
Short version: Babs realized that the crappy apartment from her mini just wasn't up to snuff, and she hasn't had time (in the past *48 hours*) to get a new place and set up her own proper system yet. So... she's commandeered the Batcave--which Dick has chosen to mothball and not use, in favor of his own lair under Wayne Tower--and the Batcomputer on a short-term basis, until she can get her own setup going again.
Oh, and she did so with Alfred's permission, too, so I'd take that as adequate approval.
(Also, the reason Batman set Cass up in her own cave was to try to get her away from Babs, who was too serious about making her a fully functional human being instead of just a Weapon for pursuing the Mission, not jealousy over who gets to play with his toys, unless it was that he saw Cass herself as one of those toys...)