My buddies and I do comp stomps all the time, it's not too bad once you get the hang of it (so long as someone doesn't mess up to bad).  Here's a couple tips for comp stomping:

1) You're humans, and you're a team, so communicate!  Coordinate your attacks and defenses.

2) Try to establish just a couple choke points where at least two of you have built a starbase.  That way if something gets messed up and you get pushed back you have a bottleneck to rally at.  Even the best AI fleets struggle with two decent starbases, and if you focus on cap ships the rest of their fleet will usually break and run.  (Starbases in general work like gang busters against the AI, max out the HP as quick as you can and let it soak up the damage while your fleets take out Cap ships and otehr high priority targets).

3) Make sure that you divide up the planets and such during the initial expansion, don't shaft your teammates by gobbling up all the good places, and then share the ones you conquer.

We didn't get creamed, but to say you were even beaten by a hard comp with even players it might as well be the same thing.

Also the map was Razors Edge, so it's close quarters. Does this change the approach? I assumed you'd still have two on offense and one economizer even on a small map.

So most of the points are correct (althought I didn't go into the detail yall did), but a lot of people disagree on 3. You can rush to illum, but you'd better start making a fleet when you get there?
Ya? or No?

Don't feel bad, I won a 3v3 PvP match that turned into a 1v2 PvP against decent players with two AI allies on my side and one AI ally on their side. My teammates dropped in the first 10 minutes. I think the first one dropped after watching our teammate in a good position to rush camp until he had three Halcyon's built (I was curious to watch and see what he would do actually). The Halcyon builder dropped after the first dropped leaving me 1v3. The third guy dropped on their team after I wiped out his dual egg and twenty assailants. He was able to take out one of my low level halcyon's though.

I would have doubts that a pure economizer is needed on a smaller map like this. It would probably be just as efficient to purchase health upgrades on planets and do the civil researches that increase planet population after fleeting up. I might have one of the rear line players put a culture center up too.

I prefer to fill my fleet spots against the AI early. That means ten or so disciples,cobalts,etc early then adding LRM's with the next fleet upgrade, then adding carrier support. I would probably mix in a second capital ship too, maybe a carrier or a siege cap later depending. The AI sometimes gets creative with it's fleet composition so be prepared to counter it.

This map is going to see a lot of early AI activity. You will need repair bays (overlapping), turrents with the repair bays to keep them engaged. If you can control the two choke point asteroids on this map you should have a distinct advantage. I would have the aggressor attack those two roids and skip the ice/volcano (no civic labs). I would have the middle player take the two adjacent ice/volcano and the asteroid adding fleet as he is able to the front lines. The third rear player should probably expand forward through the roid then the ice/volcanic and engage with the aggressor on the front. Be aggressive with the AI, it will most likely come with a larger fleet than yours so you will need fleet and defensive structures to counter that.

Try not to use the black market too, work together see if you can share resources you need. Also, try not to sell/buy resources to queue up multiple ships/research etc. Monitor when it completes you may have the needed funds by then without selling/buying. Early game this can provide a huge benefit to your economy.

You know, a team comp stomp in Diplomacy on hard I would actually be interested in, just so I could get acquainted with team play.  Is there a particular place where I find like-minded people?

TEC is just as offensive as any other race.  All races are offensive.  Don't fall into the thinking that TEC isn't offensive.