All the answers to....
What are the known laws of the village?
In the case of a woman of the village being beaten by an outsider (someone not of the village) then if that person is female then she is given the Kiss of Death and then be chased until one of them is dead. If the other person is male, then they must marry.
Would a woman from the village have to be beaten to get married?
Almost certainly not. The women are probably on average stronger than the men in the village, and so not every woman would be able to get a husband if they had to get beaten. Since these women would be the strongest, it is not very sensible to stop them from marrying and having strong children as they are a warrior race.
The village could work something like this - the women train, so that they can become as strong as possible, and then fight each other to see who's the strongest. The strongest unmarried woman would most likely want the strongest available man (they'd probably fight to prove this as well), so it effectively boils down to a pecking order, with the stronger women getting the stronger men and having (on average) stronger children. The women in this female dominated society might also marry for other reasons - ie love, arranged marriages, wealth, position etc.
It is possible that the women do have to beaten to get married, but thats rather unlikely in this matriarchal society, and it would probably also leave the strongest women unmarried. They could let themselves loose, but it would be very unlikely that they do that as it would be a terrible blow to their pride.
What would happen if a married woman got beaten by an male outsider?
Maybe marry that person as well - nothing says they can't have more than one husband. Or the two guys fight to the death, and the winner gets the woman. Or the woman commits suicide for loosing again, or something else.
What if a woman got beaten by a married male outsider?
Maybe just ignore this and keep the guy in the village. Maybe kill the other wife, or just marry the guy anyway, or something else.
Why would the Amazons have the laws Shampoo has to follow when she was beaten by Ranma?
Since the women of the village would most likely be a lot stronger than most of the other people in the world, then if they get beaten by an outsider, it probably means that person is very strong. So, if the person is a woman, she has to die a) because she humbled the Amazon and b) because it'll stop them from having strong children with out outsiders - so their village gets stronger in relation to the world. And likewise, if the person is a male, then they must marry because it'll increase the strength of the village compared to the outsiders.
It also makes a very useful plot device.
If Mousse beat up Shampoo, would Shampoo have to marry Mousse?
Almost certainly not. There are no laws stating what happens when a woman is beaten by someone of the same village. If it was the same as regarding the outsiders, then obviously the law would say it applies to everyone because it wouldn't be sensible to have it as two separate laws. So, it is very likely that men of the village don't have to marry any female they beat. Also if the law did apply to anyone, the women wouldn't be able to fight with others at all, because someone would loose, so someone would have to die, or get married every time a female has a fight.
What would Mousse have to do to get Shampoo to love him?
Cologne has said (V5.6) that the women of her village want to marry a strong man, and to Shampoo Ranma is the strongest young man in the area, and also very strong compared to everyone in the world. So, Ranma is the best available man to her, and so Mousse would have to be shown to be better than Ranma for Shampoo to consider marrying Mousse - though Shampoo does love Ranma, which'll cause some problems.
Also, he'd have to stop mistaking other objects/people for Shampoo - which pissed her off when they were children and so she rejected him, and still does because he hasn't learnt yet. If Ranma married Akane, then Mousse would have an easier time, though Shampoo would not necessarily give on Ranma immediately, but if he was the strongest bachelor in the village he would be best suited for marrying Shampoo.