I purchase one artillery unit for my mid-war German army. At this point, if I purchase nothing else, the unit becomes an on-board direct-fire gun, with no indirect capacity and no assets.
No, you can place the artillery off board, but if you do you will need to purchase an FAO or assets (or both) to use it. You can also place the gun on table but it's a lot safer for it to be off-table.
If I also purchase 3 assets, that means I can keep this unit off the table and schedule 3 artillery strikes during the game. It would not otherwise be able to fire. Right?
If you have an FAO, you can fire the gun on any turn that you are not using an asset. Essentially, if the gun is not firing using an asset as scheduled artillery, it is sort of sitting around waiting for an FAO to tell it where to fire,
If I also purchase a FAO, that means I can also make command rolls to delay or cancel my 3 assets. But I can't suddenly place the unit on the table and use it as a direct-fire weapon. Right?
Correct.
Basically, neither assets nor FAOs are mandatory when buying artillery, but if you don't buy them, your gun just sits on the table like a direct-fire piece.
[A}Neither are mandatory, but without at least one of them, off-board artillery is useless, so you might as well deploy it on board.