Old outlets and grounded indicator light?

I just moved in to a (rental) apartment in the top floor of a house built in 1900. It's been rehabbed a couple of times, but the outlets are still few and most are two-pronged. In a couple of rooms some boxed, grounded outlets were installed at some point. In one of the unboxed, presumably ungrounded, outlets, the previous tenants left in a three-prong adapter. Out of curious its, I plugged in a surge protector and the green "grounded" indicator light came on. This surprised me, but made me wonder if somebody had found a (safe?) way around this. Are these surge protector/grounded lights reliable? This one is from Ace Hardware. The outlet plate is old and painted over, but it does have a screw in the middle. When I tried to remove the adapter, there was a lot of resistance, making me wonder if it's screwed in behind the plate.

On top of it being a holiday weekend, the landlord and management company are not really on top of things, so getting an electrician in would take time. Also, the circuit breaker panel isn't in a very accessible place, and I don't really understand enough about this to try that on my own anyway.

My initial questions are simply if this outlet is grounded after all, and if I t's safe (for me and my emectronics) to plug a surge protector in, and if there's a cheap and reliable way to check on the other old outlets. The bedroom has no three-pront outlets at all, and it would be nice to find a way around that. My other question is, about how much would it cost, if I were to hire an electrician on my own, to have some of the outlets safely "converted," as well as GFCI outlets installed in the bathroom and kitchen? I live in Richmond, VA.

Apologies for the long post. As I'm sure is obvious, I understand just enough about all this to recognize how little I understand!

Thanks in advance!
-C