can i move a downloaded copy of macos installer onto an external drive from my computer

You're welcome.

I and other users still haven't figured out why Apple started removing your OS purchases. And on top of that, why only the latest ones?

Edit: Oh yeah! That's why. Older versions were tagged with the user account of the person who "purchased" it. So things such as this would happen: You install Yosemite on a Mac and then sell it. The new owner boots to the Recovery partition to reinstall the OS. Uh, oh! They can't because now it asks for the Apple ID and password of the person who installed it.

Apple stopped assigning OS purchases to each user starting with Sierra. If you want to download, or re-download Sierra or higher, you just go to the source. Such as this link for Sierra.

How to upgrade to macOS Sierra - Apple Support

Scroll down to section 4. There's a link to download Sierra, which will open the App Store. It is kind of goofy. You have to follow this link to even get Sierra to appear. Doing a search for Sierra in the App Store gives no result.

Somewhat. Once you've purchased anything on the App Store (free items are also a purchase), they remain in your list of purchased apps.

But, Apple changed how they handle what versions of the Mac OS remain in your list. I still have Lion, Mountain Lion, Mavericks, Yosemite and El Capitan in my App Store purchases list, but for whatever reason, Sierra, High Sierra and Mojave are no longer there.

So yes, I would strongly suggest saving a full installer of any OS Apple releases.

Yes.

Yes you can save a copy of installer on external drive and copy it back to Mac when you want to use it.

But you can also always get it from App Store.