Android

I believe you can do it from the Aurora Store under Manual Download. But you would have to have bought the app already as you would have to do it having signed on using your own account on it. Since you cannot purchase it via Aurora. I have done this for some icon packs and a few other aoos that already purchased.

Be aware, as I mentioned elsewhere, even if you download it, if the app constantly checks if you bought the app via the Playstore then unless you have the Playstore running and signed on as well, then it will likely fail to load/launch and will tell you that you have not paid for it. Even if you actually have, as it cannot verify it without the PS.

There are someapps that I have purchased, and that I can download but they will not work after they are installed.

Might be a split '.apk' file (aka '.apks'),
for which you can try SAI (Split APKs Installer, to backup + restore):
https://f-droid.org/packages/com.aefyr.sai.fdroid/

Might be due to a check to see if the app was installed from the PlayStore,
for which you can try KingInstaller (Spoofs as PlayStore, does not work for split APKs, to restore):
https://apt.izzysoft.de/fdroid/index/apk/com.example.kinginstaller

Might be due to custom licensing check,
then you'd need to decompile, reverse engineer and write a bypass.

Interestingly enough Files by Google has this functionality

If you can install it you can use LuckyPatcher

As far as I know you can do this only with a rooted device but I may be wrong

I thought an APK file was basically a ZIP file with a standardized structure. You should just be able to rename the extension to ZIP to open it up and view the contents.

Not like that's going to decompile the code and all, but that's the first thing I'd start with.