I did not serve a mission myself, but I know lots of people who have, so I’ll pass on a little of that info.
Yes, they pay for it themselves, and it lasts about 2 years
Yes, they spend a great deal of their time proscletyzing (sp?), but the trend now is to do it off referrals from members rather than door-to-door. That doesn’t mean they never go through neighborhoods knocking, but I think some personal safety issues have arrisen in the not too distant past that has cut that down considerably.
Once they have completed their training in the MTC in SLC(training for foreign speaking missions takes additional time), they go to an assigned area (they do not get to choose) and are attached to a stake/ward/branch (terms for groups of members that meet together – depending on how many members there are). Their assigned area may change many times during the course of their mission. Many times they may be asked to help members in their local area or provide emergency service to the community. If a member in their area has a friend that is interested in knowing more about the church they will refer them to the missionaries to help teach them about LDS beliefs and doctrine.
A mission is basically 2 years of service to the church. Missionaries don’t get much free time and have little opportunity to practice whatever it is they would normally do for enjoyment (TV, music, reading secular material, etc.). The last time I moved we tried to do it ourselves (big mistake) and when I called the Bishop of the ward we moved into he sent the Missionaries out to help us get our stuff situated … it was very helpful. Ward members often sign up to feed them dinner on a specific night (we have a calendar that gets passed around every month) because we know they are making do on their own funds and it gives us an opportunity to provide a service for them.
They live a spartan lifestyle where practically their entire time is spent in scripture study, teaching, and service to the church and it’s members. It can be an incredibly rewarding experience. Although I know of a few stories of unsuccessful missions where the kid just couldn’t take it or did something stupid and was sent home, the majority of return missionaries that I know say they reaped immeasurable benefits from those years of service.
FWIW, they are not interested in teaching those who do not want to learn, and if you tell them to go away, they will. They are not there to sell you a Book of Mormon, but they will give you one if you would like to read it. If you need help (i.e. you’re water pipes just burst in your basement or you are digging up a new garden plot) they will roll up their sleeves and assist you.
Disclaimer: I am not an authority on LDS Missions, I just gave you the best answer I could to the OP. If I left anything important out, there are other LDS members on the SDMB who could probably answer better.