Concrete – What to do with baseboard for this basement with concrete box

You would usually caulk the gap between the baseboard and the wall. This is why I usually don't bother painting the trim before putting it up (unless I'm staining it).

Usually I would prime the wall first, then put up the trim, putty/caulk, paint the trim, then finally paint the walls.

Yes, but.

There are so many issues with putting wood flooring at or below grade that I never like to see it done. The main issues are that wood swells with humidity, and humidity varies with seasons and other situations like rainfall. Most people think that their foundations are moisture-proof, but that is absolutely untrue with any masonry product, especially concrete. So the first thing you're going to need to do is a moisture content test on your concrete. These can be purchased at a hardware or flooring supply store, and you can send the test off for $10 US or so to a lab.

The next thing you have to determine is how to affix it. Around here in Texas, with slab-on-grade construction being most common and cellars being relatively uncommon, it's most likely to be affixed with construction or flooring adhesive. The main problem with affixing it this way is that it does not allow the wood to flex, expand, or contract without breaking loose from the adhesive or potentially cracking boards. I have seen some flooring installers in the Great White North (as we refer to the land you come from) who put down several inches of rigid foam insulation covered with a tongue and groove plywood product and affix the wood flooring to that. I would research that option -- I have no idea what it takes and if it's practical for your home, but I know that it would remove some of the height from your basement. With a father who's 6'4", and a best friend who's 6'7", that would be of concern to me.

The last and final thing you need to consider is what you're going to do if your basement ever floods. Wood floors are ruined by flooding. Rooms that are below grade are prone to flooding either due to rising water tables or infiltration from above, including from inside the home.

Instead of hardwood, I would strongly recommend that you look at ceramic tile that is designed to emulate hardwood flooring. It will provide the look that you're seeking without all of the problems. With an in-floor heating system, it could also provide the feel that you're going for at a lower end-cost than wood flooring.