Is there a browser extension to get all the CSS that is applied to a DOM element?

Switch to Chrome default element inspector (press F12), it has all that you need. You'll find everything in the Computed Style panel, including a useful "Show inherited" checkbox

I know the question is almost 4 years old, but if there is someone looking for it today, there's a Chrome extension that handles it. https://github.com/kdzwinel/SnappySnippet

It adds a new tab in Chrome Inspector and you just need to click a button to get all html and css of the selected element and its children. Then you can export it to codepen, jsfiddle and jsbin, or copy and paste.