Generally, flooring is tongue & groove and nails are driven at an angle through the groove and into the subfloor/joist below. If the flooring is not T&G, then it would be face nailed. If you're not seeing nails from above, then the flooring is, most likely, T&G and nailed through the grooves.
It's possible that it was glued down, but from the age of the wood shown in your other question, this was probably put together before construction adhesive was a big item, so I'd consider it an unlikely option.
To remove the joist, I'd recommend a reciprocating saw (Sawzall™ is a brand name that's become generic for the type of tool I'm talking about), and run a combination/demolition blade (designed for cutting nail embedded wood) in the joint between the bottom of the flooring and the top of the joist. You'll end up cutting some of the joist and, unfortunately, some of the flooring, but the goal is simply to cut the nails connecting the two. Err toward cutting the joist since that's going into the garbage/fireplace. Once you've cut the joist away from the flooring above, one or two vertical cuts will allow you to remove the joist in a couple of pieces. If any nails remain protruding through the floor, attempt to pull them through or cut/grind them flush with the bottom of the flooring above.
Another option would be to use the Sawzall™ to cut the joist into short 4-12" lengths, then bang them sideways with a hammer to knock them loose from whatever is holding them to the floor above. Some of your pieces will just fall out/fly away when you hit them because there doesn't happen to be a nail into this particular piece, others will take several hits and/or some prying to get them off the nail above. The more termite eaten bits may just crumble and fall apart as you hit them. These will leave their nails intact and they'll need to be dealt with.
If the flooring was actually glued down, the wood (especially the more termite eaten bits) may split and break because the glue (whether construction adhesive or a modern PVA type glue) may well be stronger than the wood itself. If this is the case, break off whatever bits you can, then Sawzall™ as noted above or chisel/scrape the remaining bits away. If this is necessary, you'll want to scrape and sand all the old glue off so that you have a pretty flat, smooth surface on the bottom of the flooring so the new joist will seat nicely from below. Chunks left here will telegraph into uneven spots in the flooring above at worst, or make installing the new joist difficult and leave gaps that may lead to squeaking floors somewhere down the line.
When installing the new joist, I'd suggested "squeak fix" clips for screwing the flooring down to the joist from below so you're not messing up the finished surface above. You could use construction adhesive on the top of the joist to attach the flooring to it. "Glue & Screw" is a common recipe for squeak-free floor installation - generally between the joists and the plywood/OSB subfloor. While you don't have a subfloor, it would work here - just bear in mind that if you ever have to repair any of the flooring from above (and it appears that this area may be in an entryway so it's at a higher risk of long-term damage from water ingress/wet shoes), gluing down the finished floor will make it quite a nightmare to pull up any damaged sections of finished floor from above.
