I appreciate the credit for how to find the third.
You have most stuff right, but you are missing a couple of hard limits that are screwing your stuff up a bit. I have not found these these limits in the perspective books I have read, for reasons that escape me.
1. The closest corner of a cube may never, ever be an angle less than 90 degrees. Unless you are distorting perspective. I use a circle to locate my vanishing points on the horizon because it defines the limit of my draw able area. 2. Vertical lines appear to be perpendicular to the horizon at the horizon. I have found a very specific method of locating the 3rd vanishing point, based on an angular distance from the horizon. Based on that method, it is possible to define vertical lines as curves, but that takes too much time. Perspective is usually limited to a small field of view because straight line perspective geometry does not accurately describe how we perceive the world, and methods incorporating curves usually look wrong and take a lot of work. Have another look at my video on 2 and 3 point perspective, and let me know if understand what I am getting at here. [link]
I agree that you should have an idea of the 3 dimensional form, and your position relative to it in mind before you start drawing. it might help to make a simple drawing of the 6 sides (front, right, back, left, top and bottom) of the object to reference when doing the perspective drawing.