How do you clean the inside?

Do not use water on wooden boxes. Salt residue has already broken down the wood fibres over the centuries. Moisture will accelerate the damage. Brush out loose dust with a dry bristle brush. If it is a ceramic antique salt box, a damp cloth is fine for wiping the glazed interior.

I bought an old wooden box that has a really strong smell of woodsmoke and old grease, is there a way to get rid of it?

Odours get trapped in the porous timber. These pieces lived next to open fires for generations, absorbing soot and cooking fats. You can place a bowl of dry bicarbonate of soda inside and close the lid for a few weeks. This gently absorbs the smells without risking damage to the carcase. Do not sand the inside to freshen it. Stripping away the historic surface ruins the patina.

Can I still use one for salt?

Yes, provided the interior is clean and stable. Ceramic or stoneware examples are ideal for modern use as they wash easily and resist corrosion. Wooden boxes are trickier. Modern table salt often falls through the dry joints. Coarse sea salt works better, but it will continue to slowly degrade the timber.

My box has a small drawer at the very bottom underneath the main slanted lid compartment, what was this actually meant to hold?

That small lower drawer held spices. Nutmeg and mace needed the same dry conditions as the salt above them. Keeping them near the hearth prevented dampness.