There is no difference between a prologue called a prologue, and one called 'chapter one'. If the information is irrelevant to the story, do not include it. If the information is relevant to the story, but only in a setting/atmospheric context, include it only when you need to establish that setting/atmosphere.

Does a prologue count as a first chapter?

Prologues come before chapter one and could be expository/introductory prose, a poem, diary letter, news clipping, or anything in between. As a reader, when I start reading a prologue, I'm usually impatient to get to chapter one.

Does a prologue count in the word count?

Sure, a prologue is part of the story and would be included in the total word count. An introduction is something else, usually written after the book has been purchased by a publisher, so it wouldn't be included in your word count.