That depends if you’re doing MIDI or audio. You could export your Music Studio project as a MIDI (.mid) file and import it in Logic. The instruments will of course be different and you would select them in Logic to your liking.
For audio tracks, or if you want to retain the sound of your MIDI tracks and export them as audio tracks, the best way is to export to WAV format and enable “Save tracks separately”: https://www.xewton.com/musicstudio/doc/?id=9#Export
I don’t quite understand what you mean, why would the start times be different if all exported audio tracks start at the beginning of the song?