No. Not if the GFCI is wired correctly. The neutral wire should always be cold, or at ground potential.
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Not if the GFCI breaker is supplying the circuit you are wanting to put the GFCI receptacle into.
Deoends on code you are governed by. In USA, a GFCI outlet or a circuit controlled by a GFCI circuit breaker would be required.
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.
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.
Normally it is a 20 amp using AWG 12/2 gauge wire. But it really depends on what size wire is on that circuit. If it is white AWG 14 gauge then use a 15 amp breaker. If it is yellow AWG 12 gauge then use a 20 amp breaker.