Do they stay lit constantly, or only when you press them?
Depends on the game/application. Usually, in a poker machine, the lights are controlled by the game, and only valid choices are illuminated
Do they require additional wiring for the bulb?
Yes. It's really just a normal switch, with a housing and holder for the light bulb. The light bulb is wired completely separately from the switch. You can power it however you want - games that are intended to use lighted buttons control the bulbs too, just like a pinball machine.
Is the writing printed on the plastic or is there a paper insert underneath that you can swap out?
On all the ones I've had, it's just a little plastic (think transparency film) insert that goes under the button. You can pop out the cover and replace the legends on the buttons. Paper works too.
Also, can these be set to a push on/push off (SPDT?) style?
Possibly, although I've never seen one like that. The lighted game buttons I've seen all use standard microswitches, like normal arcade buttons. Find a SPDT version of one of those, and you'd be in business.
-Ian