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.
It is NOT safe to obtain the grounding from another outlet.
The grounding wire must follow along side the hot and neutral wires of the same circuit for the entire path back to the panel. The reason for this is to minimize the inductance in the event of a short circuit to any grounded metal, either in work boxes, or in the powered equipment itself (such as the multi-outlet box). Surge protection will be much less effective in cases where the surge is a differential between hot/neutral and ground (which actually is a common type of surge) due to this same inductance.
Inductance is minimized when the wires with the current going in both directions are close together. Their magnetic fields overlap and mostly cancel each other out (there will be a field inside the cable between the wires). When the wires are far apart, their magnetic fields will extend a greater distance and there will be more inductance. Also, metal located inside the "loop" between the wires can be affected by the magnetics (sometimes in a hazardous way).
To correctly upgrade a 2-wire circuit to 2-wire-plus-ground circuit, the cable must be replaced with one that has integrated ground. In some rare cases, the proper cable may have been run, already, but an ungrounded outlet was used.
Simply running a single wire along side the old cable is also NOT safe. The wires of a circuit, when conducting current, will try to physically move apart from each other due to the orientation of the magnetic field. Physically binding the new single wire to the cable with cable ties not further than 6 inches apart for the entire length could avoid that issue (but is still not code compliant). If you are doing that much work, just replace the cable with the proper type. FYI, this was one of the hazards of older knob and tube wiring. Single wires inside conduit are known to move around, but the conduit sufficiently confines the movement so all you get is some noise.
In many cases you can obtain much (but not all) of the safety of grounding with the use of a GFCI outlet. The grounding wire still cannot be used in this case. It just gives a safer 2-wire load. It is not sufficiently safe for appliances that have frequent human contact, like a computer.
Best Answer
This is an obsolete socket type called a parallel and tandem socket, and is designed to accept either standard parallel plugs without a ground or older tandem style plugs (again without a ground):
None of the connections is a ground - it consists of 2 pairs of hot and neutrals, one oriented vertically, the other horizontally. If there is a ground wire to the box (which I doubt), this can be replaced with a standard 3 prong outlet. If not, you'd have to find a way to get a ground into the box.