How do you clean the interior fabric?

Brush it gently with a soft, dry bristle brush. Dust gathers in the bottom of the compartments over the years. Never use water on watered silk or old velvet. Moisture stains the fabric and dissolves the animal glue holding it to the wood. Leave stubborn marks alone. Aggressive cleaning will tear the old fabric.

My box has a lock but the key is missing, so is there a way to get a replacement key made for it today?

Yes, a good locksmith can cut a replacement. Box locks use simple warded mechanisms. You need to take the box to them so they can measure the pin and file the key blank to fit. Do not try to force modern keys into the hole.

What goes in the square compartments?

Those held the inkwells. They usually sit near the front edge. Most boxes originally came with matching glass bottles. If they are missing, the empty space is obvious.

I recently bought one with a sloped wooden front that drops down on hinges, does this mean it is actually a writing slope?

The line between the two forms is blurry. If the drop-down front rests flat to form a writing surface, it shares DNA with a lap desk. A dedicated stationery cabinet usually stands upright and focuses primarily on paper storage. Those hybrid boxes try to do both jobs at once.