Should I apply caulk before or after washing with TSP

If you must paint this stained wood, a light sanding with 4-O steel wool or 320-grit sandpaper should be fine. All you're doing is roughing up the surface coat, usually poly, which will let the paint "key" to the surface better.

Be aware of the finish originally used on the wood; only poly will take a latex topcoat well, while on most other finishes like varnish or oil-based finishes, latex will not bond to the surface properly; it may peel easily, or even bead up when applied. If this is the case, roughing up the topcoat is not enough; you will have to sand off the topcoat to get to raw wood, and on a profiled wood piece like mouldings, this can be difficult or impossible.

Also be aware of the type of wood; certain woods are "open-grain", and will absorb paint deeply into the structure of the wood, requiring many coats to "fill" the wood and get a solid color on top. Otherwise, the grain and any knots will show through the final coat until you put 3 or 4 thick coats on. If you have to go down to bare wood on an open-grained wood, you have to seal the wood with a product that is compatible with your paint base.

No two whites are the same. In fact, sometimes two tins of the same color from the same company can be different! This is why if you are painting large areas, it is best to blend the cans together.