"Example: 3:15 - fifteen minutes past three OR a quarter past three.
Do natives agree with that?"
For me, in British English, yes I do agree with that. You will often just hear, "quarter past three", or "Three fifteen", as well.
"Can we not say, 'It is fifteen minutes past three a.m.' or 'It's ten (minutes) to ten p.m.'?
It makes sense, but I would never say it, unless there was a confusion over whether the time being discussed was three in the morning or evening. Or, in your second example, ten in the morning or ten at night. This is why many people now use the 24 hour clock method of telling the time. However, it is best to try to become familiar with both. The 24 hour clock is sometimes used because the time given tells you exactly the right time of day or night. This is the reason it is often used for flight times.
However, It is more natural to say something like "three fifteen a.m.", or "three fifteen in the morning", or "Ten to ten in the evening", or "Ten to ten at night", or even, "nine fifty p.m.".
Most British English speakers wouldn't naturally use the 24 hour clock method as it seems very formal.
They would tend to use the a.m. and p.m. system of telling the time that you are learing (called "the 12 hour clock") but not often say the "a.m." or "p.m." bit, because usually it is obvious whether the subject in question will happen in the day or the night. That is why we, usually, don't write or say them unless it's necessary.
So: "We will hold the meeting at three tomorrow", in the vast majority of cases would mean "three in the afternoon".
Student or Learner