Not sure about the dropdown menu, I believe the functionality behind that does come from the Genesis Framework itself though. I’ll try it later on a tablet if I get a chance.
For the Logo, you can add this to your css –
.header-image .site-title a {
background-size: contain !important;
}Give that a try and lets see if that fixes it. It worked for me when I tried it with your site/logo using my web inspector. If it doesn’t work, then we’ll probably need to move it to a certain area of your css so it can have priority.