If you use a GFCI outlet and the rest of the outlets downstream don't have a ground wire attached, are they still considered grounded and are they safe?
They are not grounded, though may be GFCI protected. If they are wired off the load side of the GFCI, then they are GFCI protected. In which case they should be labeled with stickers that say "GFCI PROTECTED", and "NO EQUIPMENT GROUND ".
Tester101 nailed the question, so I'll cover another option.
Depending on your situation, it is often possible to retrofit ground. Normally, wires in a circuit must be kept together for good reason. Ground is a special case, it can be routed separately from the other wires in a retrofit situation.
That is because ground is not used to flow current except in a fault condition (when a breaker is about to trip). The job of the ground is to give a nice easy path for the fault current to flow back to the service panel (breaker box) and trip the breaker, without passing through any humans or setting anything on fire.
Does a GFCI make an ungrounded circuit safe? The GFCI is comparing the current on "hot" and "neutral". If they are the same, all the current is accounted for, and none of it is shocking anyone. The issue is whether the GFCI is precise enough to detect a dangerous current - this varies. 10ma is enough to knock someone out, and that is fatal in some instances (they fall, they drown). The other issues relate to equipment: RF noise causing problems, etc. The equipment may not perform well without a true ground.
No, they are not grounded.
Yes, they are GFCI protected (as long as they are on the LOAD side of the GFCI).
You are allowed to protect older ungrounded outlets by GFCI protecting them, though it's still preferable (but more invasive) to actually get ground wires everywhere. If you have any 3-prong outlets without a ground, they must be GFCI protected and labelled "No equipment ground".