Does the Chromatic Scale always (I mean always) use fingers 1-2-3?

In pianistic terms the chromatic scale is amorphous: you have to decide on any combination of all (5-)finger patterns according to context, from (RH ascending) 'efficient' 1-3-1-3-1-2-3 to (e.g. Chopin op.10/2) 4-3-4-5-3-4-3-4-5 and anything in between these extremes. So, to answer your question bluntly: No, the chromatic scale uses any that serves all the physical demands as well as all the musical ones the most efficiently in the cirstances you find yourself in.You end up with dozens & dozens of possible s, given experience, and are then able to choose from these at will and/or hybridise them even further if necessary. In the example from op.13 you quoted, the fundamental pattern is sensible: it's seeking to keep (4) on black and alternate as tightly as possible -- because of the key layout -- between that and (3) on white as you negotiate the (ef) and (bc) 'gaps' forcing a pattern change. It's not the only solution imaginable, depending on the colour you want to achieve, but it's a pretty workable & effective one.