The dream is that the “true physical theory” will give a prediction for these numbers.
In string theory, the forces are envisioned to be higher dimensional extensions to the usual 4-spacetime. Compactifying these higher dimensions to 4-spacetime produces a solution for the fundamental constants. The problem is that there are many ways to compactify higher dimensional spaces to the usual 4-spacetime, and each one generally produces a different solution for the fundamental constants. For 10 dimensional string theory, there are ~10^500 ways to do this. This is known as the string landscape.
It is still a subject of research whether we can find our universe within the string landscape. Note that while the number of solutions of physical constants (the jargon in the field is vacua) is dauntingly large, it is not infinite, which is a good thing.
No, these fundamental dimensionless physical constants are all independent of each other.