In PIM Sparse Mode, can the Rendezvous Point be the first or last hop router? In tutorials and configuration examples, usually, the Rendezvous Point is described as the central registry to match multicast senders with receivers. FHRs and LHRs send PIM messages to the RP to be matched to each other. Is it permitted for the RP router to be either the first-hop-router or the last-hop-router? That is, can a sender or receiver hang off a Rendezvous Point?

Yes, in fact, receivers do hang off the RP. When a listener joins the RPT, necessarily, the MROUTER serving that listener sends a join toward the RP. If there are some hops between listener's mrouter and the RP, those routers all join that group so they can receive traffic for the group via the RPT.

Sources also effectively hang off the RP once the RP decides a given source's data rate is high enough to warrant joining an SPT for that source, instead of relying on the source's DR to encapsulate traffic in PIM register messages.

There is no restriction on which router can be an RP. You can locate it anywhere. This is a good document from Juniper which discusses some issues related to planning your RP / multicast topology. I also highly recommend reading Interdomain Multicast Routing if you haven't already.