Quite probably it would work without greater problems but it's recommended to use the same releases on server and desktop clients. I assume that you didn't have a separate test-environment (which would be recommended for such cases - I know the budgets ...) but for testing reasons I would only switch one machine and if this runs without problems upgrading the whole environment.
Yes you can. I do this all the time with my QV Desktops at the company and at clients (at this very moment, my desktop is SR13 and my own server is SR6 - no difficulties encountered so far). But I do not easily move out to the next major release like QV12, just from SR to next SR.
Although installing QV12 Desktop shouldn't cause any conflicts with an 11.20 server. To be sure, just read the Release Notes if you plan to upgrade your Desktops to QV12.