How do the Reavers reproduce?

I don't have any official sources to cite, but my guess is that the Reavers' ships contain air from Miranda's atmosphere, and that it still carries traces of the drug. When they dock with another ship the drug in their air diffuses into that ship, and transforms anyone there with the genetic susceptibility. Of course the Reavers butcher everyone else, and to casual observers it would seem like some small percentage of people on the ship were driven mad by the carnage.

The Reavers do seem to direct all of their aggression to non-Reavers, and therefore must have the ability to recognize one another despite their berserker rage. Since they leave the newly transformed Reavers alive they must be able to recognize another Reaver even if he or she doesn't look or act like one yet.

My one nagging question from the Firefly episode was why did the attacking Reavers leave the new Reaver behind? One possible answer is they expected that when he fully transformed, he would pilot the ship back to Miranda. However there is no clue given as to how he would be able to find it. Homing instinct? Telepathy? We'll never know.

There is no evidence that Reavers reproduce or grow their numbers at all.

There is nothing in canon to indicate that the Reavers' numbers are growing at all. The recording on Miranda (linked below) indicated that the PAX drug introduced to the population turned one tenth of one percent of the colony's population of 30 million people into Reavers. That's 30,000 Reavers, which more than accounts for what we ever see on screen. Further, Reavers are somewhat uncommon as you get farther from Miranda. The characters are typically relaxed as they fly around, and settlements are not armed camps built to defend against Reaver attack--running into Reavers is a danger, but it's relatively rare. In fact, it's rare enough that Simon Tam likened Reavers to campfire stories. So 30,000 Reavers could pretty easily create the fuss in the system that we see depicted on the show and in the film.

The idea that the Reavers left the survivor from Bushwacked alive as a means of perpetuating their kind has flaw in it.

In the episode you cite--Bushwacked, the third episode of the series--Mal does say that the young man was driven mad by what he saw and predicts that he will begin to exhibit Reaver behavior, but he does not say that this is how Reavers recruit new numbers. In fact, this idea has a pretty big flaw in it: if the Reavers wanted to recruit this man, why didn't they take him with them? Surely he would have descended into madness just as well locked alone in a Reaver cargo hold for a few days as he did in the kitchen cabinet on his own ship. So why leave him there if he was a new recruit for the Reaver fleet? More likely they kept him alive and made him watch what he did to the rest of the settlers as an act of sheer cruelty and maybe as a living booby-trap for whoever came along to salvage the vessel. In fact, we saw that they booby trapped the ship in another manner, so the survivor as a ticking time bomb booby trap fits their modus operandi and the theme of the episode.