The issue you are experiencing is not a crop error.
The header is fixed to the top of the screen, so when you narrow the screen size your logo and menu cascade onto 2 rows and cause the height of the header to increase.
Also, you should disabled the “auto-height” option in the Slider Settings and set a fixed minimum height.
As for the first issue you mentioned with the horizontal scroll bars, http://www.spanknet.it/img/errore_navigation_bars.png
If you have entered HTML characters where they should not be this may be the cause.
I would suggest disabling ALL your plugins until the problem goes away then you will have identified the culprit.
Lastly, it would be much easier to debug if we had access to the site you are working on.
My IP is 18.104.22.168
Most of your issues are not bugs but configuration issues.