If you do not want to fight too much with multiple browser issues than first rule of thumb is do not break html rules.
For example in your HTML within topnav ul some items are without li parent.
I haven't seen ever anything within ul without li.So first step should be keep your screept, link and head cart within li or take it outside ul and than style it.
There also some other issues such as search button styling.
First fix that than you can look into menu issues