I have a lot of experience doing commissions- been doing them since 2005
So here's my takes on your questions!
It does depend on the person, but I've found that stuff that is colored flies best. It doesn't need to be "complete" IE: fully illustrated/painted/etc, but the people who want sketches-only tend to be more rare, or people who want commissions snail-mail sent to them. But, some people cannot afford a piece that is bigger and more complicated, so it's good to offer different tiers to suit different needs.
I don't think you'd be missing out on a big market. You should offer only things you feel confident in drawing- I've never seen an issue with people saying "i don't draw furry/mecha/backgrounds/etc" - having that listed up front helps people from asking and then getting turned down (or you having to turn them down), or otherwise suffer through a commission that you're less than confident about.
Generally, I think people list types of commissions (sketch, color, portrait, full body, icon) because it is much easier for people to decide from a list with pricing, like a menu, than it is to come up to an artist cold and ask about services and such. It's less stressful on both to know what they are getting into and what they will get out of it.
It's up to you if you want to only offer certain tiers/pricing, or only offer specific commissions on a special (only portraits only 5 spots!) or if you want to be generally open for any types of commissions. In my experience, the most important thing is to do what you, the artist, is comfortable with and that you are sure you can complete in a timely manner, and then also be sure to showcase work that you know you can meet the quality of. So I try to always show other commissions as my examples of what I can do- that way what you see is what you get. 
Best of luck!!