AE and Camera Tracker doesn't finish solving if on a tripod?

If the camera is not moving there is no need to track anything. You can use Corner Pin on a layer to position and simulate perspective. It's just like building a composite in Photoshop. No need for 3D at all.

If the camera is on a tripod the only thing you can get is a panning shot track and there is no depth data. If you do not have a lot of fixed geometry in a shot, the camera tracker will not work for the same reason that camera tracking a panning shot or a zoom shot does, not give you any depth information. You only get 3D depth information if there are significant parallax changes in the shot that can be analyzed. Anything that is moving in the shot, people, cars, waves on water, trees blowing in the wind, foul-up calculating the parallax (perspective) changes and result in tracking errors or complete failure.

Show us the shot you are having problems with or accurately describe it and we can probably tell you if it is suitable for Camera Tracking.