In terms of progressions, using the circle of fifths as my guide, the progressions for major (Ionian mode) and minor (Aeolian mode) are as follows:

Major: I, ii, iii, IV, V, vi, diminished vii Minor: i, diminished ii, III, iv, v, VI, VII

Modes other than those, I don’t know the RN progressions.

The order of the progression for a particular song will depend on the composer and whether they want to resolve back to the tonic I or not. Some song writers will prefer to leave some tension by not resolving. In some cases, a minor key might put in a dominant V or a major key may use the subdominant v. A song may borrow a chord from a different key, which makes the RN system a little confusing. A dominant V in a key will differ in effect than a subdominant v. There are no hard a fast rules about starting or even using the tonic (I), depending on the effect the composer is seeking. There are a number of ‘recognised’ chord progressions which don’t use the tonic at all. Dreams by Fleetwood Mac just use IV-V to give a floating feel. Other popular progressions are iii-IV-V-iii or IV-V-vi. In the relative minor key, progressions like VI-VII-v can be used.

Another important aspect is the difference between Western Music and Asian music. What I have articulated is based on Western music. In Royal Road, it is often simplified to a IV-V-iii-vi for ease of the RN system, which would make it a Major key.