Are they in a private house shared together (with or without servants) or in a boarding house together? I'm not sure former would be normal, but the later was quite normal. James Buchanan, the America's only bachelor president, lived 1834-1844 in a boarding house with his close friend (and, by some accounts, lover) William Rufus King. The amount of time they spent together and their particularly close relationship caused some eyebrows to be raised (or, at the very least, some jokes to be made), but their living together in a boarding house wasn't seen as particularly strange, I don't think. Buchanan's situation might be an entry point into research of how two men living together in very very roughly your time period (fifty years later) was possible.

However, I'm not sure if single men regularly lived alone without servants. There was a strong sexual division of labor in the 18th century, meaning that many of the laborious household tasks (cooking from scratch, doing laundry by hand, cleaning without a vacuum all take a considerable amount of time) were most often done by women: either family members, paid help, or a proprietress of a boarding house. This is a century later, but Sherlock Holmes and Watson famously live together in a boarding house. Horatio Alger's stories also commonly feature boarding houses, and one of my favorite novels, Pere Goriot, focuses largely on the residents of a Parisian boarding house. However, these are all 19th century examples and I can't think of any ready examples from the 18th.

Children, however, complicates the matter--I don't know whether it was at all common for men to raise children without a female presence in an urban (rather than agricultural or frontier) environment. That's not to say it didn't happen, but rather all these cases I can think of of widowers with children tend to quickly remarry--I can't give an exhaustive list, it was just always my impression and surprised me to see "wife died in year N; remarried in year N+2." Again, I always attributed this to the strong sexual division of labor. I imagine the cases where a widower didn't quickly remarry were case where he had another woman in his life--a sister, aunt, female cousin to live with him; a daughter old enough to take on the household tasks; hired help--to do the copious "women's work" required in running a household. With children, women's work was not just the physical labor of cooking, cleaning, sewing and mending, but also the emotional labor of raising children up right. They were to be, in many ways, the moral center of the household. That ideology (woman as "Angel of the House", imbued in a "culture of domesticity", head of her own "separate sphere") really developed in the mid-19th century, but I don't wonder if there were traces of it already in the late 18th, especially in conservative Boston.

I'm not positive, but I think men raising children without a woman would raise more eye brows than a man boarding with the family of a close friend. Obviously, you could make it a plot point, having other characters say, "Don't you think you should find a new mother for your children?" all the time, but that's something else to look out for in researching this era.

Not particularly.

I had a professor who would say that "nobody had a lifestyle before 1975," at least when we tried to put modern connotations on historical lives. In that sense, there wasn't a culture of gay men living together openly, as romantic partners, so men living together would not be presumed to be gay.

Finances and social status would of course, play some role in this. Like now, you don't come off as a bigshot if you're living on a friend's couch. But it may have been a practical necessity from time to time, regardless. And wealthier people had live-in staff in multiple roles; the unmarried veteran could be a tutor or something if necessary.