Ghost shrimp sold in N. America are almost always Palaemonetes paludosus. They will reproduce at room temperatures 70 to 80 degrees. I don't know where the cold water information came from, I've never read that. They are a very hardy species. Your picture is of a fully developed shrimp, so it was probably a few days old. When they first hatch, they are free swimming ( but don't look like an adult shrimp) and go through one or two larval stages. I've only been able to confirm one. Since I don't have a good microscope (or any for that matter), I can't say more than that. I've kept this species in treated tap water in So Cal, Utah, and Arizona. They are fun to watch and breed.
There have been instances where people have purchased 'Ghost Shrimp' and received one or more shrimp that grew over a few inches. These are cases where a Macrobrachium species was inadvertently included some how. They can be extremely aggressive, but usually are something like M. lamarrei lamarrei (a docile species of Macrobrachium shrimp, that grow to just under two inches). I have heard of one horror story where the shrimp turned out to be a M. rosenbergii or M. dacqueti. They are closely related species of Macrobrachium shrimp, difficult to distinguish, that are extremely aggressive and grow well over a foot long.