Afaik, windows will not install to an external drive as the sole or primary OS. It's windows after an installation that recognises an external drive being plugged in and assigns appropriate resources to deal with the external drive requirements. That can't happen if there's no OS present in the pc to begin with.
Do you need NVMe? Absolutely no, you do not. Sata SSD or HDD will work just fine, whether m.2 or standard form factor is irrelevant.
Count on windows having an average use of @ 80Gb, that includes any and all files plus any working space required by hiberfil.sys, pagefile.sys and other such requirements.
Between factory files and any extra DLC's the games or you may add onto the game, it's not hard for some games to go beyond 100Gb in size. My Skyrim + mods currently sits at 128Gb, and thats not including any files in use by Steam.
Nothing wrong with trying to cheap out on a budget, until doing so hampers the reason for the pc in the first place. You'd be far better off with a 1Tb HDD than a 120Gb SSD, even if performance from a HDD sucks, it's still better than using an external drive for used applications. External drives are for Storage/Backup, occasional use, not the daily driver.