Besides time, 零 （れい） is also used for 'degree' (i.e. rotational or temperature). And besides numbers, ゼロ is used in most situations where you have an outcome of nothing (e.g. weather visibility, someone's skills, interest rates etc.). Both of them can be used in spelling out the 0 in phone numbers and in game scores, but I can't say which is the more common one.
Your teacher is dead right, assuming you come from an Anglophone background; from the perspective of an English speaker, you try to make an r but you tap the tongue on the palette ridge, away from the teeth, basically. It will likely sound like a "d" sound, if you have never heard / said it before, but over time you will learn to not treat it as an allophone for another sound.