My primary concern would be for your safety with the circuits with GFCIs. I find it hard to believe that just those two circuits would have been upgraded to grounded cable in a remodel, and I can't see any way that a GFCI could work with ungrounded cable. Test that the GFCIs work correctly on their own. Does your receptacle tester also have a GFCI test function? If so, use it on the GFCIs too.
It's OK to have two-prong outlets if they're not grounded: see this answer to this earlier question for more information, but a surge protector simply prevents sudden changes in supply voltage from reaching the equipment it's protecting, be it on live, neutral or ground; it doesn't do anything to provide a good, stable, ground level where one didn't previously exist.
Electronic equipment often likes to have a stable ground level as a reference point. Under normal circumstances, the ground conductor in NM cable doesn't carry any current, so it's all at the same electrical potential as the ground spike outside your building. Without that stable level, a piece of equipment's internal "ground" can vary. This usually isn't a problem internally within a single piece of equipment, but can cause problems if electrical signals are shared between two pieces of equipment: mains hum in hi-fi systems would be one example of this. You can mitigate the effects by keeping related equipment close together to minimize the amount of electrical wire between them (like plugging them into the same duplex receptacle) but, depending on exactly what equipment you have, you may still run into problems.
As far as code compliance, permit applications and inspection results may be a matter of public record (they are in my locale), so you could do some research into what was done during this remodel.
- Check for unused grounding wire, and connect if found (great option, but I suspect we will not find such wiring based on the age of the building)
This is the only option that will actually provide equipment grounding, assuming you find an equipment grounding conductor in the box.
- Replace receptacle with ungrounded GFCI outlet
While this will provide ground-fault protection; which is good, it will not provide equipment grounding. In fact, when you do this you have to add a label that reads "No Equipment Ground" on each receptacle protected by the GFCI.
- Connect a ground wire from the outlet plug to it's box (presuming we discover the box is metal not plastic) and use the box for ground
This only works if the metal box is connected to an equipment grounding conductor, otherwise it offers no grounding at all.
- Ignore the lack of proper ground and cross our fingers none of our electronics get damaged
This is probably the most common solution. Though most people aren't ignoring the lack of grounding, they simply don't know they need it.
- Run an extension cord from a properly grounded outlet in a non-original part of the house to where the electronics will be?
This is a temporary solution, but you'd have to protect the cords from damage and use the proper size cord.
There's really not an easy solution here, the only way to do this is to run an equipment grounding conductor.
Other useful questions:
Best Answer
Meet grandfathering: the idea that if it was legal at the time it was built or renovated, it's still legal. However, fitting ungrounded 3-prong outlets were not legal in 1965. What else could it be?
It is legal to fit a GFCI (Europe: RCD) device. Circuits are typically wired in strings, and a GFCI device can provide GFCI protection to downline points of use. This means those are GFCI protected also. However, those points of use which are receptacles should have a "GFCI Protected" label.
If an ungrounded outlet has GFCI protection, it is legal to put a grounded receptacle there. However, the receptacle must have the "GFCI Protected" label, and it must also have a "No Equipment Ground" label.
So that settles it. It's legal if those labels (and GFCI protection) are present.
"But on this ungrounded outlet, I pushed 'GFCI test' on the outlet tester, and it didn't trip."
That is correct. If the outlet has no ground, the GFCI test should not trip. The tester can't create a ground fault if it doesn't have any access to a ground, which is kinda obvious if you think about it that way.
There are a couple ways to test for GFCI protection.
Trip GFCIs and see if you lose power here
Push "Test" on one GFCI device in the house. If this outlet loses power, then presumably it is in the protected zone of that GFCI. Repeat for each GFCI device.
Hack the tester to use an external ground
Tester into cheater; cheater into wall. At this point the tester should indicate "grounded" and the GFCI test should work.
If this fails, either the ground you are connecting to isn't a ground, or there is no GFCI protection here. Or your test rig is broken.
If it tests out OK, then contact the landlord and demand they come out and fit the mandatory labels. Then, next week, tear them off because they're ugly.
No grounds and no GFCI
This is a can of worms. The nightmare scenario is -- nah, Oakland is a rent-control city, they would never condemn the occupancy because that would make it an easy way for the landlord to evict protected tenants.
So the landlord has several options.
Many jurisdictions allow landlords to do trivial repairs, such as exchanging an outlet, switch or light fixture. All other repairs on rental properties must be done by a licensed electrician. Oakland's permit office would know whether a particular activity a) needs an electrician and b) needs a permit pulled.