Laura, there would be an issue with your drop down nav covering the top ad. And yes, I know some big sites like Huff Post have the same, but Google does not apply the same rules to big websites as it does to us.
They might say they do, but they dont, many small publishers have recieved warnings for that very thing. Also, note that with your drop down, it would be easy to accidentally click an ad below, but on the Huff Post almost impossible... Im sure that is one reason they can get away with it.
So yes, in general if a drop down covers adsense, you will probably get warned about it. Also, you do need to use responsive adsense so a mobile just shows a small banner up there, like the 320 x 100px one.
I have actually done this, put a banner at the top in the way you are talking of. Yes, it does increase the number of clicks. BUT it also reduced the price of them by a huge amount. I have gone from averaging 30c per click to about 5-10c, and the increase in the number is nowhere near enough to make up for that. I will probably be removing it soon, just giving it a fair trial of a few months first.