Electrical – Why are ungrounded GFCIs required to be labeled

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.

This is a good question which can have a variety of answers depending on the exact case. I would open up the breaker panel and see if there is a ground running to it. It is very possible that the breaker panel is grounded and that the neutral is grounded but the outlets and lights aren't. If you don't know how to tell if the panel is grounded, you could take a picture of the panel and post it here and I'll edit my answer. Many surge protectors don't work at all if they aren't properly grounded, and those that do are much less effective. When you moved in, were the outlets two prong outlets or three prong outlets? If they were two prong outlets then you can't upgrade them to three prong without making sure the whole system is grounded properly. If they were three prong outlets when you got there then you Landlord is liable and he must get it fixed. Ungrounded three prong outlets are a real hazard.

In any case, I would try to find out what is really going on and then approach your Landlord. Since you don't own the house, it is neither your responsibility nor your right to do anything major without approval from the landlord and in this case since it could is a safety issue, and it is definitely an upgrade on his house, your landlord should pay for it... or at least pitch in.

No, a GFCI will not fix the problem. A GFCI does meet code [210-7(d)(3)] and is better than nothing but it is a last resort. It is my belief that it only satisfies the NEC because they knew that some people would not do it right and they figured that the GFCI outlet was better than nothing.

Edit after Your Uptdate

You may want to make sure that they didn't use any "bootleg grounds." Those testers are easily tricked by these false grounds but these "tricks" are not safe.