There are launchers that lock apps shown on them, but they're more kids locks than actual locks.
sappy_junior728 Is it possible to set up and lock down GOS so that nothing can be downloaded from the web browser?
No. If someone downloads from the web browser, they get the popup saying they can allow installation by changing settings, as normal.
sappy_junior728 Is it possible to lock down F-Droid or any other app store I'll install with a PIN that only I'll know?
If you don't see it in F-Droid it's not there.
There are probably alternative F-Droid client apps that have locks, but I don't know any (mostly because that's not a useful feature for me).
sappy_junior728 Can I also disable side loading after the necessary apps are installed?
Side-loading is per-app, not global.
What you may be looking for is a MDM solution, but the useful ones require a subscription. HeadwindMDM is a free open-source option, but the interesting features require a subscription https://h-mdm.com/headwind-mdm-version-comparison/