Re: It is half after/past 3.
We do say "it's a quarter past" or "it's a quarter after" to mean (for example) 3:15. I use both, interchangealy.
We also say "I's a quarter to" or "it's a quarter of" to mean (for example) 2:45.
But for 3:30, it's not "half after."
It's a quarter past 3 means 3:15
It's a quarter after 3 also means 3:15.
It's half past 3 means 3:30
It's a quarter of 3 means 2:45
It's a quarter to 3 means 2:45.
We also often drop the hour.
A: I'll pick you up at 3.
B: Could you make it a quarter of? We hope to finish early today.
A: Can you pick me up at 3?
B: I probably won't be able to get there until quarter after.
A: No problem.I'll wait.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.