Hey neighbor, didn’t know you’re from Ukraine. I am from Romania. I also like a lot what you did with your website, can wait to see the live version.
I suggest you add the code to the “Header” sidebar instead of the “Full Screen” sidebar. The header sidebar has some margins that may help the items not stick to the header. Then try either setting vertical-align: top; on the image or setting display: block; on the image and a margin-top: XXpx; to the adsense block. It’s hard to tell exactly if I can’t test the code in the browser.
I’m afraid I don’t understand the question about search and 404 pages.