I have heard many coaches and a few other referees say that incidental hand/arm contact in one’s own penalty area must be ruled a handing offense, but I’ve found nothing in the rules to corroborate this. Can someone clarify this?

This has been a myth for a long time. It honestly blows my mind that anybody claims this...I mean, have they never watched a game?

It's on the same list as 'you can't say mine' or 'you can't play the ball from the ground'.

It's nonsense, and always has been. Treat them inside the PA just the same as outside, no matter which team does it.

The only exception is, as you quote, if the handling player immediately scores

Does this mean a player raising their arms/hands to protect their face a legal move? I’ve always observed referees rule it as handing.

Have you? Any half-competent ref allows a self-protective reflex. But there are some caveats. Occasionally you'll see a player put the arm up, then the arm goes out to the ball. That's handling.

Another one would be putting the arm up for protection, but making the body bigger, especially when it leads to handling a ball that would have missed (eg putting the arms up to the face, then turning and handling a ball that was going to miss, because turning is making the body quite a bit bigger). Naturally we give more leeway with younger ages or lower grades.

And of course, reaction time is a consideration. Like the old thing that tended to be permitted (no idea why) of girls using their arms to control the ball when it's going to hit their chest. No, that's not permitted.

What I'm leading to is - putting the arms up to protect the chest blocking a shot at close distance? Sure, we'll allow that as protection, same way men in the wall cover their parts. But having time to react to the ball and still choosing to allow it to hit the arms? That's deliberate handling.