I have tried several times to download a game via the browser. They failed. So, I tried the Air/Gog downloader. It completed, and I now have a file named "heroes_of_might_and_magic_3_complete_edition_part0.gogDownload" which finished downloading. Now what do I open the file with?

Try to rename the file to heroes_of_might_and_magic_3_complete_edition.exe

Fingers crossed for the file being actually complete and just without the.exe extension.

That's a fairly common problem with the GOG downloader. When it is done with a download, it is supposed to re-name the file to an executable (like "setup_heroes_of_might_and_magic_3_complete_edition.exe"), but it will sometimes "forget" to do that.

My bad, didn't see the ".gogDownload" on my first read

That did the trick... just renaming the file. The game seems buggy though (crash). Oh well, what does one expect for $7?

Check the HoMM subforum. You may find the solution for your problem there.