You need to be on the same Wi-fi as the device with the music. Then on the device without the music, go to the playlist and click the toggle for "Available Offline" This should take care of putting the songs on your other devices.
Basically no. But you can import your local files and sync them through Spotify to mobile devices.
But if you use more than one computer then you'd need your local files on all of them, for that you need to manually copy them from one to the other, or use a service such as Dropbox as hpguru has suggested. Dropbox would automate the process of keeping all the computers having the same [local] files on them.
-- First was the record, then the cassette, the CD, then came along Spotify... that really did change everything!
As long your the music is in your I-Tunes, you may have to transfer it first to their, then Go to File --> Import Playlists --> I-Tunes - And choose what you want. Also, this can be all uploaded in the very beginning when setting up the Spotify app on your computer.