The cached page still contains references to the original content (images, scripts and style-sheet files, etc.) so, when you look at the default cached page, your browser will request this content to the original website, possibly along with a "referer" header telling explicitely to the website that you are currently viewing the Google cached version of which page.
However Google offers you the alternative to see the "Text-only version" of the cached page. By this way all these tags linking to the original content will be stripped off and your browser will contact only Google servers.
And as per you second question, all your interaction with the text-only version of the cached page will remain local to your browser: you can right-click, select, copy, save or examine the source code of the whole web page or a some elements of them freely.
Foot note: the above statement is true for a "normal" behaving browser and environment. Some browser plugins and antivirus proactively check the links contained on a webpage displayed in the browser without the user having to click on them (use case can range from checking the URL reputation to actually downloading the web page to accelerate navigation). You should know if your system has got such feature installed on it. In case of doubt, a simple test with a network monitoring tool like Wireshark would tell.