A flat or sharp before the numeral applies to the root. These are adapted from figured bass conventions which name chords (or intervals or clusters) from the bass upward. Now with the basics, we can look at flats, sharps, and numbers. An augmented chord is represented by a superscript + on a major like III+ (which I don't have on my keyboard). A superscript zero (like a degree mark) to a minor chord (like ii°) means diminished. There are also the equivalent of diacritical marks to represent other things. Upper and lower case refer to typesetting machines.) The tonic (this only applies to key-centered music.) is always I in major or i in minor.

(The one's I learned on my own were from Harder's self-study books.) Major chords are indicated by capital Roman Numerals and minor chords by small letters (Spanish has better terms than English: mayuscula and minuscula. This may vary a bit from book to book, but the most use the following conventions.

So as in your examples in C, III is E-G#-B, but bIII is Eb-G-Bb iii is E-G-B but biii is Eb-Gb-Bb v is G-Bb-D but #v is G#-B-D# and #v° (or dim) is G#-B-D. Your stuff is partially right, however when the first note is changed, tho whole chord follows.

B flat major roman numerals