The most obvious reason for choosing a town barber as a narrator is a stylistic one: a barber can tell a long tale without interruption to an uninvolved listener. Another reason is that in small towns in the 1920s, the town barber shop was very often the gathering place for the local men. This puts the barber in a position of advantage because he knows everyone very well and knows all the town gossip and all the latest news and the hottest stories of exploits. In other words, the town barber is the best source of information on the male population of the town.
The question as to whether Whitey the barber is relaible or not is a complicated one. In a very real way, he is not reliable as he identifies Jim Kendall and Hod Meyers as pranksters who are full of "mischief" and fun and who have the whole laughing more than any other small town:
[Jim] and Hod Meyers used to keep this town in an uproar. I bet they was more laughin' done here than any town its size in America.
Even after Jim assaults Julie then humiliates her, when Doc Stair enters the barbershop to look for Paul Dickson, the barber still insists that Jim is basically a good man:
I said it had been a kind of a raw thing, but Jim just couldn't resist no kind of a joke, no matter how raw. I said I thought he was all right at heart, but just bubblin' over with mischief.
On the other hand, he does have an accurate perception of the other characters and he does express sympathy for the ones who take the greatest brunt of Jim's brutal "all kinds of sport": Milton, with the Adam's apple; Paul Dickson, with the fall on his head in babyhood; Julie Gregg, who rejects Jim continually.
In conclusion, Whitey's reliability is mixed: he has no right sense for the immorality of brutality, yet he has an accurate perception of decent people. This, of course, plays into Lardner's theme, which protests against condoning vicious behavior, such as Jim's and Hod's, as good old mischief and laughs: brutality is brutality no matter if cowardly people and "the rough bunch" laugh at it.
To answer both of your questions as one, I believe the reason Ring Lardner chose the barber as the narrator of the story is that he is reliable. I think a tone of reliability is set in the first two paragraphs in that you are made to feel as if the barber is talking directly to you and telling you personally the story. He invites you to get personally involved in the story he is about to tell by saying "You're a newcomer, ain't you? I thought I hadn't seen you round before. I hope you like it good enough to stay." He is inviting you to see if you want to stay and hear the story of the local people and if this is a place that will interest you. He also gains trust in that from beginning to end he never actually judges anyone. He has no problem with matter of fact letting you know of one's shortcomings but offers no personal judgment on any of them. If anything, he seems to like and make excuses for everyone. With all the horrible pranks he is about to tell you Jim did, even hurtful ones to his own family, he still claims, "He was kind of rough, but a good fella at heart." Paul Dickson "ain't crazy . . . just silly."
I also think his job itself lends to his credibility. While it is perceived that women will go to the salon to "gossip", men will do the same at the barber shop just not consider it as such. Men think they are "hanging out", playing jokes, and telling stories. Men just don't feel they are as hush, hush about it. Either way, the barber would be in the know as to what was going on with everyone in town seeing as he spends his time where people conjugate and it would be discussed. While his language lends to the fact that he isn't an educated person, he doesn't seem like one who misses anything.
Lastly, I admire how he lets you make your own decision as to whether Jim's death was done with purposeful intent. While he never claims it was anything more than an accident, and one with circumstances he would have personally avoided, he also makes sure to subtly note that Doc Stair told Paul of Jim's joke on Julie "that anybody that would do a thing like that ought not to be let live."