I know we're supposed to offer a multitude of sources in our answers here, but there happens to be a comprehensive book of such sterling quality that I feel safe in relying on it. Friend OP, I strongly suggest looking at George Chauncey's Gay New York: Gender, Urban Culture, and the Making of the Gay Male World, 1890-1940 (Basic Books, 1995). Dr. Chauncey is one of the most respected historians of sexuality in America today (having given testimony before the Supreme Court!) and this book helped cement his reputation. It would very likely answer all your questions, but just for some quick summaries:

To answer your first question, it depends on where you were and how you define 'openly' gay. Life in a city like New York would be different than in a rural area (as I'll explain in my answer to your next question), and to paraphrase, there was more than one way to be gay in the early 20th century. While some, called (often by themselves!) "fairies" were flamboyantly gay (often as part of a performance they were actually paid for, as many people in New York, gay and straight alike, flocked to performances involving crossdressers), others kept their sexual orientation as secret as they could, but still managed to maintain community ties with other gay men through the use of certain codes, subtle behaviors, etc. that went unnoticed by the larger heterosexual society. (p. 4-6)

2: Yes, in general the city was better for gay life, though not necessarily because it was easy to be "anonymous" or because there was more scrutiny on you in rural communities, as one might expect. To quote directly and succinctly from pages 133-134:

The complexity of the city's social and spatial organization made it possible for gay men to construct the multiple public identities necessary for them to participate in the gay world without losing the privileges of the straight: assuming one identity at work, another in leisure, one identity before biological kin, another with gay friends. The city, as the sociologist Robert Park observed in 1916, sustained "a mosaic of little [social] worlds, and their segregation from one another allowed men to assume a different identity in each of them, without having to reveal the full range of their identities in any one of them.

If you don't understand, essentially the argument is that city life makes it easier to put on a variety of different masks wherever you go. In a rural setting, there's not as sturdy a wall between work, leisure, and family; you go to work in the fields with your family and have fun with them afterwards. In a city, on the other hand, you go to work in an office or factory, go to a bar or theater with friends (not necessarily your coworkers), and then go home to your family. You can act differently in each of those contexts, meaning for gays it was easier to hang out with your gay friends in the theater at night before going to work as a "normal" straight-acting guy the next morning, with neither group knowing what was up with the other.

3: It could be, though not necessarily. As Chauncey repeats in the introduction to his book, "fairies" and gay/crossdressing performers were often very popular with straight audiences. And he makes the even stronger point on pages 14-20 that "gay" and "straight," as firm and mutually exclusive orientations didn't take off until later on in the 20th century--his research found that "fairies" were defined by their "effeminacy," and "masculine men" who accepted the advances of those folks were called "trades" and generally considered conventionally masculine. However, there was disapproval from many in the early part of the 20th century-talking heads in newspapers condemned "perverts," and in the 1920s and 1930s, according to Chauncey, hundreds of men were arrested for "cruising" each year. But, as Chauncey also notes, the wider society didn't actually begin to really crack down on gays until the Depression and World War II. While hundreds were arrested yearly before, after the end of WWII that number had shot up to thousands. (5-9). I hope this also answers question 6 for you--while the 'common view' often decried gay folks as "deviants" and "perverts," many "masculine" men often felt they could have sex with "feminine" men without being any less "normal," and the wider society was fascinated with male-crossdressing performers until WWII.

4: There was, though it didn't look as it did now. In fact, that we even have words like "gay" is testament to that history. As Chauncey explains, gay male enclaves in early 20th century American cities maintained the cohesion of their communities through the use of particular slang and cant that seemed inconspicuous to outsiders but relied on double entendres to pass information on to people "in the know." For instance, back then "gay" just meant happy, but if you were talking to someone who said "gay" in a certain tone, you'd, well, know what he meant (p. 18). This was actually very important, since Chauncey also notes that solid gay communities allowed their members to find housing, job opportunities, etc. in the big city, which (then as now) could be very difficult to navigate if you didn't have contacts or people willing to look out for you (p. 3)

5: Men at least were, though back in those days it wasn't that surprising. When the labor market made some sections of the city fairly sex-segregated, it made sense to room with other guys to save money on rent (sometimes in what were called "rooming houses," which were another form of communal rent-sharing living) whether or not they were "roommates." See pages 152-153.

I hope this helps, at least in regards to the gay male experience :D