Header’s CSS doesn’t work in Chrome and Opera – any ideas how to fix it?

Try this change to CSS. Replace:

.site-title a img{
    padding-bottom:8px;
    padding-top:8px;
    height:50px;
}

with:

.site-title a img {
    padding-bottom: 8px;
    padding-top: 8px;
    height: 50px;
    position: relative;
    z-index: 9;
}

li.menu-item-1924 {
    margin-left: 130px;
}

Explanation

Adding margin-left: 130px; to first menu item, will move the menu, visually, to the right, away from GMNG image. The image, and the menu, will still be overlapping, so z-index: 9; will put the image above the menu. For z-index to work, position: relative; is required.