Zillions of people, including many who post regularly to this branch, are experienced at bicycling in Cuba. Late October to mid November can be just a tad iffy in terms of hurricane (they have occurred in that time period), but I wouldn't worry too much about that, since at most the wind and rain would keep you off the bike for a couple of day.
Three weeks is a good time frame for one end of the island or the other. If you're flying into Havana or Varadero, you can easily get all the way to the western tip of the island, through or not through the Rosario mountains as you please, and then down to the Bay of Pigs and Trinidad, through the Escambray Mountains or not, as you please. And from Trinidad, either return to wherever you flew into by Viazul bus, or bike back via a different route from the one you came in on. Playa Larga and Playa Giron, on the Bay of Pigs, are both good for snorkeling. And of course, there's no shortage of music in Havana and Trinidad.
If you fly into the eastern end of the island (Holguin or Santiago) most of the routes there will take you through the Sierra Maestre and Sierra Cristal/Baracoa Mountains. Although you can loop the whole eastern end in two weeks, three weeks would be better, as it would allow you time to do the music scene in Santiago and some snorkeling off Guardalavaca or Don Lino, beaches no more than half a day's ride north of Holguin.
Any kind of bike will do. My partner and I cycled the coast of Cuba, I on a new mountain bike, he on a ten-year-old road bike, generally on good pavement but occasionally on some rough back roads. Both bikes were just fine. The most important issues are (1) good Kevlar linings for your tires, because there's a lot on Cuban roads, urban and rural, that can puncture a tire; (2) the bike itself be should in good condition because you'll find next to nothing by way of replacement parts for anything that goes wrong. (3) whatever tools or spare parts you think you might need, take them along, because you won't find much in Cuba. Ditto power bars and powdered energy drinks.
There aren't many roads in Cuba I didn't enjoy cycling, apart from the autopista and Carretera Central down the middle of the country. It is hot, boring, has precious little scenery and, especially on the Carretera Central part, heavy with diesel-belching trucks. I suggest you take a look at BICYCLING CUBA and Lonely Planet's CYCLING CUBA. The latter is out of print, but available used, on-line and in many libraries. Both guides are long out of date in terms of lodging along the routes, currency, etc. But they're still good for general route info, as most of the routes listed in either book are little changed from what they were when the books were researched more than a decade ago. BICYCLING CUBA has some good pictures of various highways, while CYCLING CUBA includes altitude graphs for routes that involve mountainous terrain. There are also several websites that include useful info about routes taken by those riders.
