I installed TortoiseGit and it's great, but there's a thing I really need. When someone else pushes something, how am I supposed to know that the remote version has changed and thus I have to perform a pull? The folder / file icon remains the green check
TortoiseGit only reflects the status of a local repository in the overlay icons (and since the number of overlay icons is limited, that won't change).
A repository monitor is planned, but not yet implemented (as of 1.8.16). Up to that you need to watch a remote repository yourself (e.g., by watching it on github/gitlab or subscribing to devel mailing lists).