In the lectures Tesla gave from 1891-1893 he explains in quite some detail the experiments he did that led him to believe that there is this second medium. As for the density, I don't know how he determined it, he only supplies the outcome. And finally the changing from longitudinal to transverse is a thought that Tesla played with (he mentions it in the above mentioned lectures). Normally a gas will only pass longitudinal waves and a solid only transverse. But who is to say that this remains the same over the entire frequency range. It makes sense to assume that at a certain point the gas molecules can no longer keep up and the gas starts to behave like a solid allowing only for transverse waves.