Your stuff is partially right, however when the first note is changed, tho whole chord follows. 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.
This may vary a bit from book to book, but the most use the following conventions. (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. 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. There are also the equivalent of diacritical marks to represent other things. A superscript zero (like a degree mark) to a minor chord (like ii°) means diminished. An augmented chord is represented by a superscript + on a major like III+ (which I don't have on my keyboard).
Now with the basics, we can look at flats, sharps, and numbers. These are adapted from figured bass conventions which name chords (or intervals or clusters) from the bass upward. A flat or sharp before the numeral applies to the root. So in C, iii is an E minor chord, III is an E major chord, bIII is an Eb major chord, biii is a minor Eb chord. This pretty flexible in that all 12 notes in any key can be represented in Roman Numerals with a few symbols: #IV is an F# major in C (major or minor).
Additional stuff that may be useful. By adapting from figured bass conventions, the first inversion of a chord may be indicated by a suffixed 63 (a superscript 6 with a subscript 3; I can't type this so I'll indicate suffixed from the top down.) As the quality of the third is indicated by the Roman Numeral, the can be skipped. One just writes I for C-E-G (in C major) rather than C53. The advantage is that one can write I6 to represent E-G-C in that order (the I means a C chord and the 3 means in 63 position). This also give the ability to write seventh chords (though there are conventions here too). A G7 chord is easily seen to be V7 and a d minor sevenths is d7. The convention is that the "7" superscript is always a minor seventh over the root (it's the most common and this saves ink and time). The most common chords and inversions are easily shown with few symbols. The quality of other than the root can be indicated by a # or b before a number. A G7 with flattened fifth (in C) is just V7b5 or the raised fifth version V7#5.
Many common patterns are easily written (I do this in my own transcriptions of pop songs.) I-vi-ii-V7 or i-iv-VII-III-VI-ii0-V7 or i-iv6-VII-III6-VI-ii06-V7 (cycles of fifths with alternating first inversion chords.) A couple of versions of the Romanesca (Pachelbel's canon) I-V-vi-iii-IV-I-IV-V or variants I-V6-vi-I64-IV-I6-ii6-V7 (stepwise bass). i-VII-VI-V (Hit the Road Jack) etc.