OK, first thing ... you have an 'onload' event on the body of your page doing a window.clipboardData.clearData(), which is throwing an error every 20 milliseconds. You should probably fix that. There are cleaner ways to try and prevent people from copying.
The problem with the date selector overlaying the input box is in one of your CSS files, where it sets width="auto" on text inputs. Not sure which file it is, as it's been run the JCH Optimize, so everything is compressed into a single file.
Not sure what the issue with the icon is, but again, it's an issue with the CSS. If you can also point me at the page where this problem doesn't occur, I'll see if I can tell.
Just out of interest, try turning JCH Optimize off and see if that changes anything. I've sometimes seen issues arise because CSS content gets loaded in a different order once optimized.
-- hugh