Is it all right for me to use the elliptical 30-45 minutes every day? Or do I need to take days off for it to be more productive?

Just fine

It's fine

Depends on how difficult the workout is. If your workout is about equal to a walk you can (and should) do them daily. If its pretty intense you will want to have days off (or easy days in between).

You should be fine to use it every day, its just a cardio exercise.
You might get muscle fatigue if you don't take a break when your body needs it though. Listen to your body and take a day off if you need it.
I would start slow, with half this time every other day. Build up the time until you are going for the full length of time you want, and then start doing smaller workouts on the off days.