Tiny pop up window with a bar loading up randomly appears. Symbol: Grey house with a circle in it

Solution 1:

This specific window is actually the macOS application verification process, which is slightly different than Gatekeeper.

Here's some more info which may help describe what it's doing:

Apple's explanation of Gatekeeper can be found here: https://support.apple.com/en-us/HT202491

A high-level overview of problems with Catalina's application verification process can be found here: https://appletoolbox.com/why-is-macos-catalina-verifying-applications-before-i-can-open-them/

Catalina has had issues with this verification process, and random problems are not uncommon: MacOS Catalina Verifying app at bootstrap

Regarding the process / application named "Backup & Sync from Google" - I've personally never heard of that, and after a very quick search online, it seemed that all roads seemed to point to an app which should be called "Google Drive" I would probably investigate to make sure it's actually a legitimate application from Google.

The icon is the same icon you'll find in System Preferences under "Security & Privacy" - it's supposed to be a combination padlock wheel on a house: