EDIT: This is all speaking in reference to playing in C major. Forgot to mention that whoops

I'm probably gonna be using terminology a bit wrong as I'm self taught on this stuff

Basically I've been learning about functional harmony, and it's been a really useful tool. I know the basics

Tonic: I, iii, vi Subdominant: ii IV Dominant: V, vii(dim)

My first question is do these funtions apply if the chord is changed? Is Cm still tonic, is E Major still tonic? What about more funky stuff like Csus4, Cdim, Edim, etc. You get the idea.

Second question, do these funtions apply to non diatonic chords?

Been messing around with this progression of I-bIII-V I've just kinda assumed the flat 3 serves a sub or dominant funtion, as it's clearly not tonic (could be wrong lol) but I have no idea. If they do have funtions, what, and those funtions don't change based on chord type, then would that mean those 5 non diatonic chords can have simple labelled funtions?

Lastly, how does funtional harmony change with modes? I read that minor modes are the same, but what about the other 6, or harmonic minor?

Any advice or info is appreciated :)