Is there some key equation the detects if a ray or sphere intersects with a plane?

You should try Google with Collision Detection Ray Face. There is a lot of website for this collision( the most classic test ). For a sphere, it is the same that Ray plane, you just have to create a tempory face with all new vertices=last vertices+Radius.Normal
It is not really difficult.

Trivial :

Check the distance of the ray/plane to the center of the sphere.
If that distance is less than the radius of the sphere then you have intersection.

Point to ray/plane distance is left as an exercise to the reader (hint: dot product)