Hey, first off, welcome to TalkBass!
For your first question, I assume you're talking about tuning your bass with an electronic tuner. In that case, the open strings are OK for that, but make sure that your bass is properly intonated: Tune each string using open notes and then do the same with the notes at 12th fret. The tuner should give you exactly the same results.
For the second, it's OK to leave your bass tuned. In fact, it should be like that because constantly tuning and detuning isn't good for your neck (I mean, your bass').
Let us know whenever you have more questions!