What you "heard" is incorrect, but Harper's advice is wise just the same.
You can paint the outer cover of the panel as long as you don't obscure any critical labels. Use a good primer and several thin topcoats to prevent runs and avoid filling stamped text. I'd probably remove the plastic latch, mask the opening from behind, and reinstall it later for a ...
Isopropyl alcohol, "rubbing alcohol", will remove most "permanent" markers (e.g. Sharpie brand markers), but may be ineffective on ballpoint pen marks, depending on the ink formulation. For this purpose, don't bother using anything less pure than the commonly available 91%, with 99% being preferred. At 70% concentration it's commonly not effective.
You might try some mineral spirits rubbed lightly on the surface but remember, there's a reason they're called permanent markers. Your choice of paint would not be good. You won't get the smooth surface you want and that paint won't hold up well on a desktop, it will scrape off easily. Your best bet would be to get some high quality spray paint recommended ...
I have fixed paint similar to this in my current house. There were large areas where I could peel it off back to the plaster by hand or with a scraper.
Your builder's first coat has failed to adhere to the plaster and now it's peeling off. This could have been caused by:
Painting with full strength paint rather than a mist coat
Painting before the plaster ...
Here's how you get paint to stick to things.
Paint cannot bond to a shiny surface; don't take my word on it, try to paint a mirror and see what happens.
Paint wants to see a surface that looks like Swiss Alps on a microscopic level - lots of jaggies and crevasses for paint to flow into and bond in shear. To human eyes, that doesn't look ...
Fortunately there are not many things that are soluble in both oil and water.
If it's a water stain use an oil based undercoat, or a shellac based sealer.
If it's an oil, grease, or wax, stain use a water based undercoat.
Use two coats of undercoat to get a good seal.