Is there any way to get a log of all the recently installed files and their respective folder paths and source directories?
Installers that ask for your login password can put a lot of crap in your filesystem.

You could create a user account in the future with standard credentials, not admin, purely for games. This maintains a separation and improves security.

But for now let's explore a bit.

To see the files created after a specific date, like when you installed the game, use find from a Terminal like

find / -type f -newerct 2018-01-01


Or whatever date you want. You can find files of a certain size, whatever, find is powerful but slow. It will, however check the entire filesystem if necessary. Replacing the slash with the tilde (~) constrains searches to your home folder. Some searches need to be run with sudo. Try man find from a Terminal.

You should also set up folder actions. on key folders. I do this on all of my machines via right click on

/Library/LaunchAgents
/System/Library/LaunchAgents
/Users/username/Library/LaunchAgents

To be alerted if installers drop files there in the future. There are corresponding LaunchDaemon folders alongside them.

When running a legitimate package .pkg installer you can generally do a Get Info (⌘I) to see a list of what will be installed.