I believe that DirectX is a requirement for running 3D Gamestudio,
but I haven't managed to find out if it is also necessary for running
a game made with 3D Gamestudio. I would guess it is, but perhaps
a game would run with OpenGL or something else? Does a computer have
to have DirectX to run a 3D Gamestudio game?

Yes.

To expand Logan's reply, you'll need the version of DirectX that the game is made in.

Before 6.30, you need DirectX 8.1. After, you need 9.0c.

So my recommendation is to just install the 9.0c runtime.

Here's the link for it:
http://www.microsoft.com/downloads/details.aspx?FamilyId=0A9B6820-BFBB-4799-9908-D418CDEAC197&displaylang=en