And how come the greatest computer expert DC has is using Bruce's computer instead of her own?
Because apparently she's had no money since before they moved to Platinum Flats and then there was the blowing up of her Clocktower Systems business/base of operations. (Which is crap, because really, there is no way that someone with Babs's internet savvy and technical capabilities can't have created some kind of user-generated content site that rakes in millions. I like to joke that she and Catwoman collaborate on a LOLcat site.)
Which is why Oracle: the Cure had her living in a crappy apartment that didn't even have a properly accessible bathroom.
Will she go ahead and move into Wayne Manor next?
See, that's the weirdest part about the whole thing. Dick temporarily relocating out of Wayne Manor makes some sense (I would imagine it's arguably part of grieving, in that every inch of the place is a reminder that Bruce is dead). But as much as Babs would be able to feel at home, chilling in front of the Batcomputer, it is still a little weird. Especially given that we have yet to see her react at all to Bruce's death, which I also find profoundly irritating. It's as if they had no friendship at all.
It also seems a little pointless, given that she will eventually leave when Bruce comes back. (And it also interferes with Dick's plans to empty the place entirely. There seems to be no acknowledgement of her setting up camp there anywhere outside of Batgirl.)