Instead of lacquer, I'd brush high gloss varnish (not poly) for a built up, handrubbed finish; I'd also use a shellac sealer coat between the stain & the varnish. I've done several pieces of furniture by the method & have been very pleased with the results. FWIW, I'm in Richmond where Biggs furniture was made (& where they had a high end antique shop) and several decades ago, I was taken on an extended tour of the antique shop & restoration workrooms by one of the owners. Quite educational.

1. What should I use to strip the original finish > Use a chemical stripper containing Methylene Chloride in a well-ventilated area. Use protective equipment such as gloves, apron, and goggles. An organic cartridge respirator would also be a good idea if you have any doubts about the ventilation. 2. Once the finish is gone, I am assuming it is okay to sand out all of the nicks in the wood? > It's hard to tell from the photos, but the top may be veneer or solid woods. Harden, I think, uses a lot of solids, but most mfrs' table tops are veneered to retain flatness. If it's solid, you are safe to sand. If it is veneer, then, no, only do a light sanding with 220 grit. Consider the dents character marks or "distressing" that you did not have to pay extra for. 3. Do you think this is Walnut? How do I match the finish to the rest of the items? Should I take a chair into Sherwin Williams to have them match the stain? > Walnut does just fine without stain. In fact, stains can muddy the figure and make it look less attractive. 4. What type of protective finish should I use after staining? I am thinking it may have originally been shellac (finish is very cloudy in some areas) > It is most likely lacquer, the most common commercial finish used in the last 60 years. Lacquer can degrade over time and light exposure (like all finishes) and be more prone to blush. What I would do: - Apply two light coats orange shellac. This will do two things 1. Bring out a lot of the luster (chatoyance) in the wood. 2. Seal in any potential contamination from silicone oil (Pledge polish any time over its life) - Apply several coats of a wipe on varnish. A readily available source would be General Finishes' Arm-R-Seal. That is a wiping polyurethane. If you want a more high-brow finish, you could use Waterlox. A non-poly choice would be Sherwin Williams' Fast Dry Varnish. You can brush either of the latter two with 5-10% thinning or wipe with 50% thinning (equal amounts of varnish and mineral spirits).

I like the color of the coffee table it is a nice relief from allthe dark wood. As for the nesting tables I would leave them and put a nice lamp there since there is lower level of light anywhere

I quite like what you have, and I also like its current finish. It has character that will translate nicely into a traditional style, unlike the other table pictured, which has a generic look to me.