I've noticed that in some communities being a non-Jew in the synagogue seems to be discouraged. In response to a request to visit I've heard responses as direct as "we don't have a conversion program and can't assist you in your journey", to the more polite "any JEW is welcome to come to our synagogue". I fully understand protecting the community and wanting to keep the creepers out but at the same time it's a mainstream belief of Judaism that benai Noach have a place in the world to come, therefore Gentiles have no need to convert. One, if I have no need to convert then why am I not welcomed as I am, aka not-a-Jew? Two, if I'm not welcomed, where the heck am I supposed to go?
Let me respond as a convert and someone who has seen a few things that might be relevant to explaining the situation.
First of all, most synagogues I've attended are fine with visitors who are not Jewish, but some synagogues have had unfortunate experiences that may have colored their view. For example, when I was in college, I went to services at the nearby synagogue and found that attending services was a pastor from a local church who, after services and outside, did try to "preach to the Jews." In Denver (some 20 years ago, I'd guess), a husband-wife Christian missionary team were specially trained to infiltrate a modern Orthodox synagogue under the guise of being willing converts to Judaism. Their goal was to fraudulently obtain conversion documents so they could make aliyah and set up a church in Israel aimed at missionizing Jews.
Sometimes at a synagogue I attended in Washington, D.C., we'd get groups of high school kids, either from churches or from reform synagogues, who came to sight-see the Orthodox Jews, like we were animals in a zoo.
But, when the non-Jewish members of my family, and other families of converts, have visited my current synagogue for special occassions, the congregation has gone out of their way to make them feel welcome.
A better experience was at the bar mitzvah of an Israeli officer studying at the U.S. National War College. He invited his non-American classmates, which included a Syrian Army officer who was surprised about how many prayers there are in the Jewish prayerbook that ask for peace. He even reported this inforamtion back to his country.
Jews are human; we sometimes know what to do, and often do not. And when we are struggling to decide an appropriate response to new people, we can fall short on proper etiquette.
I would suggest that non-Jews make their first visit to a synagogue with a Jewish friend, especially someone from that congregation. Otherwise, non-Jewish visitors should introduce themselves after services (or just before), to anyone and everyone, and let them know why you are visiting, what you hope your visit accomplishes, and what questions you have. That will break a lot of ice.
Still, any specific synagogue may not be the one for you. They all have different personalities as decided by their membership. Since, as a non-Jew, you can travel on the Sabbath, you can look around for the best answer for your place of prayer.
Jewish communities certainly exist who would welcome your attendance without any intention of conversion. You need to find one that will, but that may require some persistence. The trial-and-error process might be frustrating, and you could be hampered by what communities are in your area, but I know of both Orthodox and non-Orthodox communities that are welcoming to non-Jews. Just don't expect a welcoming attitude in ALL communities or you will be disappointed.
In answer to part 2, there are places of worship of B'nei Noach. Vendel Jones started a whole movement for them in Texas, and I believe it has spread. Some of them even have Rabbis of repute who give them guidance.
In answer to part 1, it is true that you do not need to convert, and Judaism does not actively proselytize - it is even discouraged. However, that does not mean that a non-Jew is a Jew. There are some communities that have ceased the practice of accepting converts at all for various reasons, but many synagogues are willing to accept to their services those interested in converting.
the Mishna in Celim says:
לפנים מן החומה, מקודש מהן--שאוכלין שם קודשים קלים, ומעשר שני. הר הבית, מקודש ממנו--שאין זבים וזבות נידות ויולדות נכנסין לשם. החיל, מקודש ממנו--שאין גויים וטמא מת נכנסין לשם. עזרת הנשים, מקודשת ממנו--שאין טבול יום נכנס לשם...
so non - jews allowed to entered Har Habait, allthough certain jews were not. Har Hbait is holyer then synagogue so I think it is a "kal Vahomer"
"כי ביתי בית תפילה יקרא לכל העמים" (my house shall be called house of prayres for all nations).