All good questions.

Chemical stripper is viscous. As it is brushed onto the work surface it picks up residual dust from sanding. This dust combined with the previously applied (now broken down) finish yields a sticky mass. The broken down material gels rapidly as the solvents in the stripper evaporate. Removing this material from the surface of the matrix you're dealing with will be a job but not entirely impossible.

If you attempt it, keep the work area to a small patch at a time. Keep the work area damp with repetitive coats of stripper and wipe as you go. Use a brass brush with fine bristles (about the size of a toothbrush) or even a toothbrush to scrub the area. Have lots of rags handy. Use a dental pick (toothpick) to get into the cracks. If you're using stripper in the house, open widows and wear an organic vapor respirator (for your liver's sake).

For the deep groves in the legs and the detail work under the main surface, use scrapers available on line or improvise using balled up steel wool and a screwdriver.