Looks like you just need to type the shortcut, followed by a space, followed by the search term(s), e.g. "g falkon browser" to search google for "falkon browser" or "w pizza" to look up pizza on wikipedia.

Tested using Falkon 3.0.0, Qt 5.9.5 on ubuntu 18.04.

No, "g " seems to only give access to the limited mobile-phone version of Google.
Falkon seems unable to provide use of the "Search by image", unable to provide access to the Google web cache; much of the versatile function of google seems to be unavilable if you use Falkon.

Well, Falkcon's QtWebEngine browser engine probably isn't as sophisticated as Firefox's or Chrome's, so some sites may provide reduced functionality. Some use the browser's User Agent string to identify the browser's capabilities. You can try spoofing other browsers in Falkon's Preferences under Other > User Agent Manager.

You can also configure the urls used for those search shortcuts by clicking the gear in the bottom-right corner of the search bar drop-down, or in Falkon's Preferences under Other > Search Engines Manager. Looks like the default for Google is just "https://www.google.com/search?client=falkon&q=%s" (replacing the %s with the search terms).