First the grits. You went further than you need to at 400, but it would only be too far if you were applying stain. Danish isn’t heavy on pigments, so it doesn’t matter as much. Whether you stop at 180 or 240 is generally immaterial. With either one, a finish that forms a film like varnish, shellac and lacquer have sufficient solids that those scratches will get filled. Three coats should be sufficient, but when asked how many to use, I just say keep going until you like what you see.
Rubbing out your finish would serve no purpose, since you don’t currently have anything to rub. I don’t bother with rubbing out a finish unless I’m going for absolute perfection. For that I use a 6” orbital polisher and wet sand to whatever sheen I want. Around 1500 grit is a good satin sheen (approximately 35).
Properly applied, any of the film finishes will give you a good result. I prefer lacquer since I have a HVLP turbine and pressure pot, and lacquer dries so quickly, I can get several coats on in a day. Spraying also eliminates any issues with brush marks.
Now, as to your specific issue, I’m concerned that you thinned your varnish with mineral spirits. These days mineral spirits are so dumbed-down in an attempt to be eco-friendly that it’s useless. I assume from your username that you’re in CA, so that might be all you can get. But, if you can find paint thinner, naphtha or turpentine, that’s the solvent to be using, not mineral spirits.
Frankly, if you use a good varnish, you don’t need to thin it anyway. Rather than wiping it on with a cloth, go with a good quality foam brush. It’ll be less likely to leave brush marks.
It’s hard to tell from the photo, but there’s a good chance you can just apply varnish over what you’ve got. Go for at least three coats. Arm-R-Seal is one of the best out there. And when you store it between jobs, you need to deal with the oxygen in the can. Since varnish is reactive, oxygen will shorten its life. I use Bloxygen which is just an inert gas that won’t react with the finish and is heavier than air, so it lays down on the surface to block oxygen from reaching it—hence the name. Don’t be fooled by suggestions to store the can upside down. There’s still oxygen in there and it accomplishes nothing.
-- Half of what we read or hear about finishing is right. We just don’t know which half! — Bob Flexner