Upvote:2
Yes. Spouses of US citizens and permanent residents are one of the categories of people who are exempt from the geographic COVID-19 entry bans to the US. See for example the text of the ban for UK and Ireland.
Airlines should be aware of all the exceptions to the ban. Obviously the knowledge of individual airline staff members may vary, but the exceptions for relatives have been there since the beginning of the ban (it's not new), and spouses is the most obvious category of relatives that have exceptions, so it's very unlikely they would be unaware of it.
Yes, you would need to bring proof of your relationship to your wife, the US citizen (a marriage certificate should do), and proof of her US citizenship.