My two cents:
1 - If I have Ultimarc 360s, is there any reason for a stand-alone 4-way stick?
I've read many posts and articles stating that 4-way joysticks are required for 4-way games like Donkey Kong or PacMan, because those games don't comprehend diagonal inputs. Therefore it is said when you attempt to move Mario up the ladder and you inadvertently trigger the right and up switches simultaneously, Mario responds by not moving at all.
However, I have two 8-way Ultimate Joysticks (the cheap ones) and I've not noticed this to be the case. I have played many long sessions of Donkey Kong (it's one of my favorites) and have never suffered any unwanted stops when moving the stick diagonally.
So in my case, there was certainly no reason for a stand alone 4-way stick. And I'm glad. The less clutter on the panel, the better.
2 - why would I need 6 or 7 buttons? What am I missing if I keep it to 4 or 6 buttons?
Fighting games. Mainly the Capcom titles and the Mortal Kombats. Not much apart from that other than Defender, as nitz pointed out.
If you plan on ever using emulators other than MAME, you'll want to have at least six or even eight buttons. Think about the SNES controller with L+R and start/select buttons in addition to X,A,B,Y.
3 - Is a spinner really necessary? What games does a spinner open up for me that I can't play with a joystick or trackball?
Purists will want to include the spinner. Economists will not.
I can say that I enjoy Arkanoid and Tempest just fine using my trackball. To me, it's not worth putting a spinner on the panel for only two games. In all other regards (frontend navigation), everything a spinner can do a trackball handles nicely.
4 - Is there a minimum width I can get away with? I have seen layouts at 34 inches, but am hoping to go a little narrower without losing comfort.
I have two joysticks, each with eight buttons, and a trackball in my panel. My panel's width is about 24". I think it's ample room, and my guests haven't complained of discomfort standing close to each other.
34" seems excessive to me -- It's a control panel, not wings.