Is there any tool that allows you to automatically test page under different browser and make sure CSS looks good?
P.S. I know it sounds impossible, but maybe there are some solutions (like taking screenshots and comparing them)
https://browserlab.adobe.com/ - adobe id is needed, but it's free.
http://netrenderer.com - just for IE.
Both only take screenshots, but with a very little delay.
also for ie http://www.my-debugbar.com/wiki/IETester/HomePage - It's still in alpha state so it can only be trusted to a point.
Personally though, I pretty much have every browser in my comp so I can test things out in the actual browser. Plus all of the major ones do have web dev tools so.. if you can download all the browsers that's a good choice.
And I really just test all around functionality. For example if text is little different in some browsers then so be it, as long as it doesn't get cut or anything.
Tried BrowserStack? http://www.browserstack.com/
There was some other site I occasionally used to use that returned a set of images, but I can't remember what it was, and it wasn't always accurate.
You can use http://browsershots.org/ to create screenshots on all types of devices and browsers. This might be what you are looking for.