Americans often use after instead of past in telling the time, for example, ten after six.

But do Americans say half after ... in telling the time?
Oddly, we don't say "half after" but we do say "quarter after" and "quater of" or "quater to." (The latter interchangeable spoken, but "quarter to" in writing."

"Half past" isn't used very often.

I know it's very idiomatic, but you'd be more likely to hear "I'll be there about quarter after, maybe 6:30 if the traffic is bad." Or even "I'll be there about 6:30, maybe quarter of, if the traffic is bad." In a situation like that, where it's clear from context that you mean "quarter to 7:00," you'd leave that off in speech."
Good examples GG. I never thought about those much, but being idiomatic certainly plays a big role!
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No. Half past six or six-thirty are the only normal options.
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Dear Mrs. Kindhart (or Martha):
Hoping so much to see you,

Sincerely (or affectionately),
Jane Toplofty.
Emily Post (1873–1960). Etiquette. 1922.