https://numericaltank.sjtu.edu.cn/three-body/three-body.htm shows that there are three dimensional solutions.

We have no observed examples in nature of three body equilibrium. But then again, all places we have looked are either influenced by the chaotic orbits around them of the Solar System, our surrounding galaxy, or nearby galaxies in a cluster.

There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.

Yes. Because 3 points are coplanar, so every "3D problem" with 3 objects can be turned into a 2D problem on the correspondent plane

(of course in real life your plane would keep changing, and probably some other complicated math I can't think right now)