Your question is a complex and interesting one - when we study popular culture (the origin of your question), we must be careful when we discern what is stereotype and what is a perhaps more true representation of actual phenomenon. I apologize in advance if I skirt a bit too much on the edge, but I think I have a worthwhile contribution as I just finished up a section on The Great Gatsby and Fitzgerald with a heavy emphasis on the historical context (I am both a history and English teacher) and my answer will revolve mostly around that.

The short answer is that yes, it was a lot like how you describe it. It is also represented this way in a lot of academia (see, for example, the biography on Fitzgerald in Norton Anthology of American Literature). The Great Gatsby has perhaps somewhat of a unique position here, it is a depiction of the 20's from the 20's that also of course takes on a certain amount of artistic freedom, but it also, as you say, reproduces this image of the 20's, cements it, almost, for our later enjoyment.

Flappers were the result of an emerging liberation of women: They got the vote, so they also started trying to enjoy other things in life after having received this empowerment. Sports, cars, smoking, casual sex, alcohol and riding bicycles are all examples of activities that led to an outrage at the time. Most outraged were perhaps the suffragettes (men at the time probably did not complain about better access to sexual activities), that were not too happy about fighting for the right to vote only to see women, in their eyes, make fool out of themselves. Victorian ideals of behavior (stillness, dignity, hard work) are gone, and with them, the fashion (no more corsets but instead straight dresses that do not show off any feminine shapes in particular). The 20's were for sure a time for youth, and the name itself implies it: like young birds, they are flapping their new wings, not yet flying, but trying.

Alcohol consumption changed with prohibition as well. You can get drunk on smaller amounts of hard liquor than beer, so it was easier and more profitable to smuggle. There is a lower risk of getting caught if you are drinking in your own home but we also saw the rise of the speakeasy during this period. It is tightly connected to the rise of organized crime as the demand for illicit alcohol was so large, bigger organisations were more successful. This phenomenon was touched on a lot less in The Great Gatsby, but was a staple of later popular culture such as the many gangster movies. In 1924 when the book came out, the crime families were perhaps not yet so prolific as to make a mark for Fitzgerald, and might explain why it is missing from his depiction.

Capitalism did prosper, but as usual the rich got a lot richer and the situation for the working man and woman (women also entered the work force en masse after WWI and as a result of the previously mentioned empowerment) remained largely unchanged. Consumer goods became more commonplace thanks to advances in mass production in the wake of the assembly line production of the T-Ford, so the quality of life increased somewhat.

As far as the monikers of the decade go, 'The Jazz Age' is at least a lot more true than 'The Roaring Twenties'. Jazz became hugely popular because of the urbanized black populations spreading it, and while white artists mostly co-opted it, black artists also saw some fame. You could also listen to the radio (a recent invention) and through that enjoy new and different styles of music.

Finally then, were the 20's actually roaring? It's an interesting question, and one that has seen a lot of debate, but I really don't have time to examine it further here and now, but I would like to leave you with some tips for further reading that I personally enjoyed, as I have not directly sourced things in my reply, again, I apologize but I am currently at work :)

Lizabeth Cohen (Making a New Deal and Encountering Mass Culture At The Grassroots)

Leuchtenburg's The perils of prosperity

For specifically the Harlem Renaissance, there is Huggins' Harlem Renaissance

And then, finally, besides The Great Gatsby, I recommend you read Langston Hughes (Let America be America again and The Negro speaks of Rivers), Virginia Woolf (Orlando or A Room of One's Own are great feminist works) and perhaps The Wasteland by T. S. Elliot.