For some reason, I get a line directly above the mobile menu button.
http://screencast.com/t/TXu95Vkz0fo
I added an icon, but that doesn't seem to be the cause. The line appears whether I'm using an icon, text, or both. I've set all of the borders I can find to None and the background to transparent. The color above the line in the screenshot is padding.

Any ideas on how to get rid of this line?

Hard to tell without seeing the actual page, but could it be a shadow?
Check the settings under borders and corners.

Paintbrush is right, happened to me a while ago,, the shadow default has a 1 px for some reason, but normally the default colour is white, so you dont see it until it goes onto a coloured background.

As far as your other question I can't seem to emulate the problem you are experiencing, so can't really help with that one

Thank you! It was a shadow! I thought I had found all the border/shadow options, but I missed one.