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A couple times a month.

- Is it natural to omit "of" in conversational English?

- When native speakers use "a couple" do they most often mean exactly "two" or just "a few" would you say?

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The British English version would always require 'of'. It's often spoken as 'A couple o'times a month' as we are lazy when speaking, but it'd not be written that way.

'Couple' usually means two, but it's perfectly normal to use it to mean 'a small number' and people would generally work out through the context whether it was a definite 'two things' or just a few items.

Comments  

You hear it without "of" in the US, but it is non-standard in my opinion. It is also annoying in my opinion because it sounds lazy.

"A couple" means "two", but if it turns out there are one or two more, sue me already. I would say it means the low end of "a few". A few is three or more but not many, but if it turns out there are only two, or there are seven, talk to my lawyer. These are loose terms.

There are contexts where "a couple of" is specific, but they are rare. A couple of oxen pulling a cart is a brace of them, for example.