Can anyone go to a synagogue on Sabbath to observe (in a respectful and quiet way, of course)? Do you need permission first?
Yep. Anyone can go. Wearing a kippah is a general sign of respect (just like taking off a hat elsewhere) while wearing a tallit is specifically a Jewish obligation. Also a good idea to let them know that you're not Jewish if they offer you an honor.
Why isn't Sabbath observance a Noahidic law?
Good question. I can't tell you why, but I can take a few educated guesses. When you look at the Noahide laws, they're generally humane laws without so much as a tinge of ritual to them. You don't see Shabbos there, but then again you don't see taharat hamishpachah (family purity) either. The noahide laws aren't offering a religious system. They're offering the bare bones for a humane society.
The actual mitzvah to make Shabbos holy is found in Exodus 20:8, far after Noach, and the categories of work related to Shabbat found in the gemara are related to the construction of the mishkan (tabernacle), also later.
Also, Shabbos is not just related to the seventh day of the week, but to the Exodus from Egypt as well, which, again, happens later.
But again, I'm not sure of any "official" answer, if there is such a thing.
Were Ruth and Rahab converts?
Yep. Both.
Hope that helps.
Dauer