I am happy that you have become inspired by Judaism and have become observant. May Hashem also provide you with the right person at the right time to build your own home with, in which you can share your inspiration with guests of your own.
Your questions are very insightful and appropriate.
The source for the washing you mention, called "mayim acharonim" in Hebrew, is based on the verse, "You shall sanctify yourselves and be sanctified" (Lev. 20). Our Sages (Berachot 53b) understood that this double mention of sanctity refers to washing the hands before and after meals. Two reasons are given for the need to wash after the meal: 1] The hands must be cleaned of food before bentching, in deference to the blessing. 2] Salt from Sodom used during the meal must be washed from the hands to guard one from touching the eyes and damaging them (Chullin 105 a,b).
Since we usually eat with utensils and not with our hands, and since searing Sodom salt is hardly found among us, some authorities are of the opinion that "mayim acharonim" is no longer required. Others argue that we still often eat with our hands (consider oily French fries or juicy barbecue ribs), and that some Sodom salt might still make it to your corn-on-the-cob. Further, they posit, even if Sodom salt isn't around, regular salt in your eyes may not be the healthiest thing either. Therefore, they maintain that "mayim acharonim" is still required. Both opinions are mentioned in halacha (see Shulchan Aruch, Orach Chaim 181) and both have become accepted by different communities. That's why some people do it, and some don't.
Is there any reason why men should wash and women not? Aside from the well-known fact that women are neater eaters than men (joke, no offense guys), there may be a halachic distinction:
It's true that bentching is a mitzvah for women too. After all, it is a positive commandment that is not bound by time. For this reason, many authorities consider the obligation for women to be of Torah origin. However, many others consider it to be only rabbinic, since the obligation to bentch mentioned in the Torah is juxtaposed with inheriting the land of Israel, which was done primarily by the male leaders of the Tribes (see Sh. Ar., Or. Ch. 186).
This is another case, as above, where there is not a definitive resolution. One practical difference as to whether the obligation for women to bentch is of Torah or rabbinic origin is as follows: If a woman ate a proper meal of bread and is in doubt as to whether she bentched or not, if her requirement is from the Torah, she must bentch again. But if it is only rabbinic, she should not on account of the doubt risk saying the blessings unnecessarily (ibid.).
This line of reasoning may be used to answer your last question as to why the custom developed among some people that men wash "mayim acharonim" but women don't. As explained above, it is not clear that there is an obligation for anyone nowadays, men or women, to wash after the meals. However, if the requirement to bentch is from the Torah, as with men, there may be more reason to be strict and wash. However, if the requirement is only rabbinic, as with women according to many opinions, there may be more room for leniency, since anyway many hold that "mayim acharonim" is no longer applicable.
That being said, it's worth mentioning that great halachic authorities have accepted the opinion of the kabbalists that everyone should wash "mayim acharonim" for reasons other than those given above. May we sanctify ourselves and be sanctified!
