Are they happy to be on a mission? Well I think it depends on the person I suppose. There is an expectation that most young people will serve a mission but it isn't a requirement. They aren't all American, they come from all over the world. For some it's a great experience to be involved in a new culture for others it's a miserable experience where they are homesick and don't get much success in converting people.
They have to fund their own missions, men serve 2 years and women 18 months. They have extremely strict rules and always have a companion with them of the same sex. They are not allowed to be alone with anyone of the opposite sex. They have to be up at 630 everyday and back in their homes but 9pm every night. They can't swim for example. The men wear shirts and ties everyday except their preparation day which is the day they have "off" to do Laundry, do shopping etc and often play sports etc.
However, some love it, others hate it, no one is made to do it, although I imagine they do feel a lot of pressure to do so.