Teams do not have to name a kicker until the time of the kick. If a player feels they won't be able to kick, a different player can step forward to do it.
Once the kicks go into 'sudden death' all eligible players must eventually take a kick before any player can take a second kick. If they refused at the point they must take a kick, they would eventually be cautioned for delaying the game or unsporting behaviour, and the kick is counted as missed.
If the kicker is penalised for an offence committed after the referee has signalled for the kick to be taken, that kick is recorded as missed and the kicker is cautioned
The goalkeeper can be replaced by
any eligible player
any player excluded to balance numbers
any substitute, if they still had permitted substitutes remaining
So any sensible referee will ask the team to send forward a replacement, and proceed with the kicks.
If the goalkeeper refused to take a kick when required, the same conditions apply as above.