I live in Huddersfield, I’d say most folk pronounce it ‘Haitch’, but drop the the ‘H’ at the beginning…
It’s certainly died out over the years, most non-standard UK accents have been eroded in a similar way over time. I personally think it’s a bit sad. Find some proper locals who’ve been here for a few decades and you’ll still hear it.
No. Like I said above, “aitch” is a word found in the dictionary. With that spelling, it’s pronunciation ought to be fairly unequivocal.
Just because something is “in the dictionary” (which one?) doesn’t make it correct. It may indicate a standard (in certain contexts), that’s all. There is no absolute right or wrong when dealing with language, there is only appropriate usage in context.