Can anyone offer a quick primer on this topic? Is it enough (or even worth the effort) to go to the gate of "Air X" with a ticket on "Air Y" and ask if Air X will accept the ticket. If yes, do you need to get Air Y to endorse the ticket first? What incentive/disincentive does Air Y have to do this?

Does the fare basis booked on Air Y matter much? I would guess that if Air X would rather fill an empty seat, even at a significant discount, rather than allow a lost revenue mile.

I would be willing to approach Air X, for example, if my discount Air Y ticket is one-stop, but Air X is a non-stop.

Any thoughts? Any rules or restrictions? Is there an accepted procedure?

Yes, it's worth the effort. As long as you have the time to mosey on over.
Generally the ticket needs to be endorsed by the issuing carrier over to the other carrier you want to use. This is usually no hassle if you are on a full fare (like Y or F class) or even one way reduced coach fares that don't require advance notice (like 3 or 7 days) for changes.
Getting into advanced booking/ticketing fares gets a bit more restricted and may not be allowed changes - unless of course you are willing to pay the applicable penalty for doing so.
The key here is to have the ticket "endorsed" over, because just walking over to another airline does not necessarily keep it a negotiable instrument. Here's where travelers might see the wisdom in paper tix instead of the e-tickets. I'm certainly not against e-tickets but the paper is handy in a tight spot where time is a real constraint if you have to quickly get it done from one airline to another.
Good luck, Misstree.

This can never be done with an e-ticket and can almost always be done with a paper ticket.

I handle travel arrangements (I'm not a TA) for 4 business travelers. Upon my suggestion, they have been successful at least 6 or 8 times doing what you describe. They have never had a full fare ticket when doing so, and in fact, most often their ticket was booked in the lowest fare class.

Only once were they refused when there was availability. The traveler was turned away by a terminal counter employee (told they wouldn't take it without paying $400+ fare difference), so upon my suggestion, he went directly to the gate (immediately after being turned away at the counter). The gate agent put his name on the standby list and he cleared standby and was issued a boarding pass, with no additional payment.

It is also important for me to say that in not a single instance has a ticket ever had to be endorsed by the ticketed carrier. In each instance, the traveler simply handed over his ticket to the agent who would then issue him a boarding pass. In our case, this has always been on the return portion of the trip and with a paper ticket.

As to QuietLion's comment, I'm not so sure about that. CO and HP are interlined with e-tickets (and possibly other carriers are as well), so I would suspect that it's possible that this scenario could work between those two carriers or any other two carriers similarly situated.

Perhaps "NEVER" was a less than ideal choice of words for someone who is unusally adept with the use of such!

I did NOT know that about CO and HP and I stand corrected. Still, without specific knowledge of an interline e-ticket agreement (which me may be seeing more and more of) I would always get my ticket converted to paper before attempting to perform this feat.