Not if the GFCI breaker is supplying the circuit you are wanting to put the GFCI receptacle into.
You need a GFCI outlet at any location that is within 6' of a water source. You also need a GFCI outlet in a room with a concrete floor, any garage, and any location outside the home or under the home in the crawl space. A GFCI outlet protects you from electrical shock near water or moisture. You can protect more than one outlet with 1 GFCI outlet. Connect the incoming power to the LINE side of the GFCI outlet and all the other outlets getting power from that outlet to the LOAD side of the GFCI outlet. That way they will all be protected by 1 GFCI. A GFCI breaker is used to protect an entire circuit and not just individual receptacles. It is often cheaper to use GFCI receptacles than a breaker, especially if "piggy-backed" such as described above. It is also more convenient to reset a GFCI receptacle than to reset a breaker. But your question is "why." From this I suspect you may be misunderstanding the difference between a breaker and GFCI protection. To keep things simple let me say that a GFCI does not work on the same principles as a standard breaker. It provides a much safer protection than a standard breaker. Even with a ground you need GFCI protection as listed above.
There are tow places to put a Ground Fault Circuit Interrupter. There is a GFCI breaker which would be installed in a breaker box and a GFCI outlet that can be installed anywhere. Most GFCI outlets allow you to connect regular outlets to the GFCI and those outlets will also be protected.
Deoends on code you are governed by. In USA, a GFCI outlet or a circuit controlled by a GFCI circuit breaker would be required.
First, check the circuit breakers; make sure they are all ON. Find out if any other outlets on the same breaker are working; it's always possible that a breaker is faulty. If the breaker is on and everything else on the circuit is working, it could be as simple as a poor connection inside the outlet. Also even though the outlet is not a GFCI, it may be fed from a GFCI outlet. Check near by outlets to see if any are GFCI and are tripped.
If it's a GFCI receptacle and the button is not resetting then change the GFCI outlet.
Your question is a bit vague, but let's try a two part answer. If you have a GFCI breaker in an electric panel you should only have one connection at the breaker, but the breaker will protect all devices on the circuit. If you are talking about a GFCI outlet, they are equipped to extend the GFCI protection to other non-GFCI outlets by using the proper "output" connection on the GFCI.
GFCI Breakers are quite a bit more expensive than a GFCI outlet. More often than not a typical residence will need only a handful of GFCI outlets that combined together will be cheaper than a GFCI breaker. If you need to protect a series of outlets with GFCI protection you can simply connect the rest of the outlets on that same circuit downstream from the first outlet on the line and make that the GFCI. All you have to do is connect all the other outlets to the LOAD side of the GFCI outlet. If a GFCI fault occurs in any of the outlets down stream they will trip that very first GFCI plug you placed and keep you safe.
It is a Ground Fault Circuit Interrupter or GFCI. It can either be equipped in your electric panel as a GFCI breaker, or in a GFCI outlet which also lets you extend the GFCI protection to other outlets "down the line" from the GFCI outlet.
Circuit breaker or GFCI outlet with local reset button.
No, reduce the load below the ampere rating of the circuit breaker, or increase the wire size and breaker size to match the existing load
It's always dangerous to have a 3 prong outlet that isn't grounded unless it's a GFCI outlet or electrically down stream from a GFCI outlet. Down stream means the outlet is further away from the circuit breaker -- not physically further away, but further from the breaker electrically