I don't have a source to back me up here, but IIRC, Magic Carpet used texture-mapped polygons.

Maybe fun trivia: James Schmalz trying to replicate Magic Carpet in pure assembly language and showing it to Tim Sweeney resulted in the embryo of UnrealEd and what later became the Unreal engine.

No, it's a polygon renderer of some sort. But it is quite restricted. Each "world" is a sphere, and checking again in a youtube video, it looks to me like the sphere has a regular tessellation. It's dynamic though: some of the game mechanics raise or level hills. I'd guess they found a way to special case a lot of the rasterization given the spherical restriction. Kinda like how Descent based everything off intersections of a unit cube.

I don't know, but based on a gameplay video it looks more like vector graphics to me. The terrain silhouette seems to have straight lines.

I don't know how they pulled that terrain rendering off with 1994 hardware. The limited rendering distance probably helps.