There aren't really any specific rules that I know of about this sort of thing, because in reality it shouldn't happen (I'm not saying it doesn't happen, just that it shouldn't) - if a missionary is focusing on the work and obeying the other missionary rules (like staying with your mission companion, not being alone with members of the opposite sex during your mission), then this sort of thing shouldn't occur. You're supposed to "lose yourself" in the work and be totally focused on serving and teaching. 
 
Anyway, missionaries don't date or to parties or stuff like that - the mission is a time to work and to serve. If they can't obey those rules, then they'd need to go home. So in this sort of a situation, the missionary would probably be moved to a different area to work and told not to contact that person again (at least not until after the mission). The specifics of what would happen probably depend on the circumstances though. 
 
As far as leaving a mission, you always have your "agency" or freedom to choose, so if you want to quit, obviously nobody can force you to stay. People are generally discouraged from quitting their missions for superficial reasons like homesickness or because it's really hard work, but mostly because there's a good chance they'll later regret that decision. People do sometimes end their missions early due to health reasons or a death in the family (although often they just go home temporarily and then return to finish).

Missionaries have a whole slew of rules of behavior that they must follow while on their missions. Most of them are for the purpose of keeping them safe, keeping them out of trouble, keeping them focused on the missionary work and protecting them from false accusations of misconduct. 
 
Regarding the specific examples you gave, there are no rules against falling in love. There are only rules about how you behave and interact with others during your time as a missionary. In the case of the second scenario where a someone professes love to a missionary, in general they should politely explain that their duty as a missionary prevents them from returning any such ovations until after their time of service as a missionary is complete 
 
There is an exception for Sister missionaries, however. For the young men in the church, their primary responsibility is to prepare themselves to serve a mission when they are 19 and then, once they have returned, they are encouraged to find themselves a wife and start a family. The young womens' primary responsibility is to prepare themselves for temple marriage. If they reach the age of 21 and do not have any immediate prospects for marriage, they are welcome and encouraged to serve missions, but, if during the course of their mission the opportunity arises for them to get married (i.e. a missionary they were waiting for finishes his mission), they are welcome and encouraged to end their mission early to go home and get married. 
 
For the young men, there is no such exception. I think one of the important purposes of the mission is to prepare the young men for marriage. Missions teach important lessons in co-habitation, selfless service, personal and companionship study and prayer. I guess the young men need this much more than the young women. 
 
If a missionary decides to leave their mission early other than for the reason described above or for medical or phychological issues, every effort will be made to encourage them to finish out their time but in the end, the choice is theirs. They will probably feel stigmatized a bit for a time at least. They will probably have regrets later for not sticking it out. They won't be kicked out of the church, however, and they will still be loved and fellowshiped.