What to do with an apartment with only 2-pronged outlets
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.