I personally like HAPP Super 8-Ways (AKA Supers). They're pretty good for all games (I'd hate having a stick that's made for fighters and use it for puzzlers and have it feel weird). I have them in my cabinet and they work very well for all games I've used them with (fighters, MD3, Metal Slug, NTM, etc). JHendrix recommended them to me back when I first was getting the cabinet and I couldn't be happier with them. It uses a round actuator with the microswitches with the little lever on it (I think they're called "activated microswitches" but I'm not sure). I might add that the stick has a bit of a long shaft on it before the "teardrop" and they're not "bat-style" like most normal sticks (which are almost all grip and little shaft); another factor in getting them was that I liked that this setup was kinda like what a candy cab uses (but a candy has a less-thick shaft and a small ball instead of a teardrop).
I've used Competitions in arcades before, they get the job done, but IMO supers are better. Still, it seems that most arcades use Competitions and they certainly are good sticks (some say they excel at fighters because of the square actuator), so for the real arcade feel I'd go with these beefy sticks. They're "Bat-style" and that's what a lot of cabinets have.
I think I've used 360s (at the Stargate, I need to ask the owner), and they're nice, but IMO I like hearing the clicks of microswitches to know I'm doing something, know what I mean? These are "shaft and teardrop" style.
As for Ultimates, when I got my cab it had ultimates in it, and they were JUNKED. That says something about their quality. I've used it in arcades I think (well that or a Competition in bad condition), and they're BARELY passable. You can do better; I'd even be hesitant to take them for free. They're "bat-style".
As for buttons, well I personally have convex buttons in my cab (I _think_ they're Competition buttons) because I originally wanted to use candycab style buttons but couldn't (they're convex too), and they're great buttons. The concave style buttons (Ultimates, I think, but I'm not sure) are good too. The concave buttons allow for a better grip of your fingers on the button (it kinda traps them), and if you're going for the "true arcade feel" then I'd get the concave buttons because that's what most arcade cabinets use. But convex buttons are still good, and I personally can use either no problem.
For sticks, I'd personally go with a HAPP Super 8-Way, with convex buttons, but that's just my opinion. Since you want the real arcade feeling, then you should probably get Competition sticks with concave buttons.