I have three questions:
I am having an argument with my mom’s HOA about who is responsible for this repair. Links to authoritative sources would be much appreciated, especially for the second question.
That is "brick molding" and it is normally part of the door assembly. The main door, not the storm door (which may be attached to it, but a prehung exterior door virtually always has brickmold [alternate term for it] as part of the assembly.) Here's a link to a manufacturer page (no affiliation, no endorsement implied.) https://lakewashingtonwindows.com/front-door-parts-diagram/
It has practically nothing to do with bricks (unless your house is brick), but that's what it's called. It's made of wood or nowadays plastic. It is not made of brick (well, you knew it was wood, but just to be clear, despite the confusing name....)
For that to be rotting, it's very likely that the flashing (to divert water away from the door) is not done correctly.
That looks like a trim component. It is not structural. It also looks rough and splitting with pealing paint but not necessarily rotted. It could be sanded, filled and repainted with a good exterior paint, or replaced with an exterior non wood trim product that would never need painted. Weather it is the HOA's responsibility depends on many factors, so we need more info. Is it a Condo or a PUD with HOA? There are distinctly different rules for each.
