There's nothing wrong with having a standard 20A GFCI on a 15A circuit (provided the socket itself on the GFCI is a 15A socket, and not a 15/20A combo with a T-shaped slot). The 20A rating is just a maximum. On the other hand, having a 15A GFCI on a 20A circuit is not right.
One thing to help troubleshooting is leaving the load connections disconnected temporarily. If the GFCI won't reset under those circumstances, either you have the line connections wrong somehow or the GFCI is faulty or you have something with a ground fault plugged into it. If it works there, but not with the load wires connected, you have a ground fault somewhere downstream, and the GFCI is apparently working properly.