Mari--
I've just done research on a lot of the same questions. There's a good website with info on lasers
hereThe type of laser can matter, primarily depending on your skin color and hair color. If you have pale skin and dark hair, most any type of laser should give you some kind of result. From what I've found, most people have the best luck with a diode laser. The place I've picked to do my sessions at use a LightSheer (if it's good enough for Kate, it's good enough for me

)
Newer lasers are probably better in the sense that they're newer, but as new technology develops some of those lasers (like IPL) have not been proven effective at any kind of long term hair reduction.
I can't answer your question about regrowth. It's only been a few months on my test patches and they look great, but I have no idea what I'll look like in a few years. My first appointment for my full face is in two and a half weeks.
Regarding your other question about spacing your appointments--
the goal with repeat sessions is to get hairs during the growth phase. Not all hairs grow all the time. If you pluck your eyebrows or get them waxed, you will notice that some grow back very fast, in less than a week, while others might take a month or so to grow back. The ones that grow back right away are still in growth phase, while the ones that take longer are dormant and become active later on. The laser, from what I'm told, is most effective on hair follicles that are in the growth phase, though I don't actually know why that is.
The standard recommendation for repeat visits is usually 4 to 8 weeks, averaging 6 weeks. This gives the dormant hair follicles time to be activated, and should be long enough for regrowth from a hair follicle that wasn't sufficiently damaged with the first treatment. If you pay for two visits, and then have to wait for six months until you can afford some more laser, then you may need an extra treatment at the end to catch follicles that were active during your "off-season" but have maybe gone dormant again. This is how I've had it explained to me. I don't know that this will necessarily equal more treatments, but I suppose if your treatments are sporadic enough it could. I mean, it's possible that you might miss the same follicle(s) during an active phase on more than one occasion if you have to take a long enough period of time in between treatments.
I would definitely recommend getting a test patch done. Some offices do it for free. If they charge, it might be cheaper just to have them do your whole face. A test patch can be $100, and a session might only be $400. For the money, it's better just to get the surface area covered. Anyway, you can see how you'll react to the laser. My test spots, for example have shown very little regrowth, and some of them had been zapped a few months ago, so even two laser sessions could potentially produce a very big reduction in facial hair without much regrowth.