Ananth,
You are correct that RSTP requires P2P links for it to function. However, RSTP shouldn’t run on a portfast link, since the a portfast link should be turned only only on end-point devices and therefore shouldn’t have BPDUs on the link.
A switchport being classified as point-to-point vs shared is independent of its being classified as edge vs non-edge. Here’s a quick definition of each, since this might be useful:
Point-to-Point: A port that is operating in full duplex mode
Shared: A port that is operating in half duplex mode
Edge: A port that does not connect to any other “infrastructure” device, such as a switch. BPDUs should not be received on an Edge port.
Non-Edge: A port that connects to other switches or devices that send BPDUs.
Given the definitions above, in a properly functioning RSTP environment should have inter-switch connections classified as Point-to-Point Non-Edge ports.
The difference between a designated port (DP) and a root port (RP) has to do with which port is considered “closer” to the root bridge. A switch uses its root port as the path towards the root bridge, whereas the other side of the root port is the neighbor switch’s designated port. Almost always RPs and DPs are on different sides of the same link. Ports on the root bridge are always marked as Designated Ports