Let's look at your options...
Brisket - the gold standard of BBQ beef. Becomes very tender when cooked low and slow. An inexpensive cut of meat (if you know the right places) Cons: Very small window between undercooked (to tenderness) and dried out. Only fat is in the fat cap, meaning that you lose lots of rendered drippings every time the meat is turned. Not very spit-friendly.
Rib Roast - Good marbling throughout. Not much connective tissue to break down in the meat. Cooks relatively quickly, even at lower temps. Delicious. Cons: If you have to cook to a more "done" temperature depending on your audience, the meat could become tough and dry.
Eye of Round - Cheap. Fits on a rotisserie stake perfectly. Cooks evenly throughout. Cons: even a good round roast has a more dry, tough texture.
Chuck - Tender and flavorful. Very forgiving. Excellent for low/slow cooking. Lots of marbling throughout the meat to keep it moist during cooking. Cons: Falls apart when cooked properly, meaning it would be difficult to keep it on a spit. Best served as pulled beef, which may not be what you're seeking.
Sirloin - Juicy, full of beefy flavor. Lean. Cons: it is easy to turn out a piece of sirloin that is like shoe leather.
Tenderloin - Tender, and tasty. Forgives cooking beyond medium if done properly. Cons: Very expensive.
Beef back ribs - Tender, good fat content. Good for slow cooking. Cons: can be difficult to serve/eat.
I don't really know what I'd pick here. I'd almost certainly eliminate brisket, eye of round, and sirloin, for various reasons, if I'm sticking to spit-cooking. I'd be on the fence with chuck -- you'd have to come up with some kind of cage-type mechanism to keep the chuck roast(s) together. So I'd lean toward rib roasts or tenderloins. Both are pricey, but they are worth it. Only problem is, if you have people who insist upon medium-well or greater meat, you'll have to figure out what to do. Maybe cook to medium and sear off chunks for the people who want theirs more cooked?
Another intriguing option might be beef back ribs. I saw an episode of Primal Grill with Steven Raichlen where he made giant beef ribs on an open fire, Argentinian style. He put the ribs on big stakes and pounded them into the ground next to the fire, turning them around every so often.
As far as wood goes, I'd suggest oak and/or hickory. Personally, I like those with beef. Perhaps the best strategy would be to build a fire with charcoal, just some regular briquettes (no Match Light!!!). Then continuously feed some wood logs to the fire during the cooking process. The smoke flavor you impart will not be as profound as it would be in a smoker, but that may be appropriate to your audience.
I'd suggest getting small cuts of each of the meats, put them all on the same spit, and give it a test in your backyard. Whatever you don't like, toss it into some stew or chili. Through trial and error, you can get your fire method right, and get some good meat in the process. Win-win!
Eye of round can be difficult to cook. A roast tends to produce the best results, but it can be cooked in steak form as well. There are a couple of directions you can go.
First is to cook it as a steak, but do not take it past Medium Rare. Marinate before cooking. Slice it thinly after cooking, across the grain. Serve with sauce. Think London Broil.
If you like your steak more done, then I wouldn't recommend cooking it as a steak. Instead, make Swiss steak out of it. Or better yet, country fried steak (yeah, forget I said Swiss steak). If you pound/macerate the meat, it will be pretty well tenderized, and you will get a good meal out of it.
Best Answer
The best way to test if it's done is to use an instant-read thermometer. Slide it into the thickest part, away from any bones. I like to lift up the meat with tongs and poke it into the edge, all the way to the center. You're looking for about 135-140F for medium rare; take it off a little early if anything because the temperature will go up a bit once it's off the heat. Let it rest for ten minutes or so before digging in to let the juices re-distribute.
I don't like using nonstick for steak, or olive oil. Neither does well at high heats, which is what you need to cook off moisture at the surface and get a nice browned outside before the inside gets overcooked. Cast iron and steel (stainless or carbon) both work very well in my experience. For fat, use a high smoke point oil like safflower oil, or clarified butter.