Since you don't have a crawlspace/basement below, or an attic directly above, you're going to have to do more investigating to figure out which way your joists run.
If you're going to remove that post, you're going to have to do some drywall patching on the ceiling. Since you're going to have to patch, you may as well cut into the ceiling to find out what's going on up there.
The easiest way is to cut the drywall back to the ceiling joists so you can attach the patch to the joists above. (There are other methods, but this is the easiest, IMHO).
Buy yourself a drywall (or jab) saw like this:
Image supplied by Lowes.com, no endorsement of the vendor or the brand intended or implied
Before starting this operation, be sure to read the cautionary foot note!
Punch it up into the ceiling near the post (use the base of your hand or a hammer) and start cutting*. I'd recommend cutting in the Family-Room-to-Kitchen direction, as that's the direction I believe you will find floor joists above. If you go more than 8", stop, return to your starting point, and start cutting in the opposite direction. You should run into a floor joist in less than 6" going in this direction. Normal floor joists will be on 16" centers which leaves about 14-1/2" inches between them. You should run into a floor joist somewhere within that 14-1/2".
If, for some reason, you don't run into a joist, go back to your starting point and start cutting in the Back-yard-to-Front-yard direction. Again you should find a joist within 14-1/2".
If you still don't find a joist, it's possible that your floor joists are on 24" centers meaning that there is 22-1/2" between them. This is highly unlikely, but it's possible. Extend your cuts until you find that wood!
Now you know where your joists are. You will, most likely, find that the post is attached to the side of the joist above, though it could be directly under a joist. Again, since this is the only post, even if it's supporting some weight now, it's highly unlikely that it is structural, since it would only be supporting one single joist, not a whole series of them.
Using the same jab saw, punch up into the ceiling near the post. Cut a small square opening big enough for your cell phone to stick through. Once you've got the square cut out, turn on you phone's camera & flash (you'll most likely have to set it to video mode to turn the flash on and have it stay on). Stick your phone up into the ceiling and have a look around.
You're looking for:
- Joist locations - which way to they run, how far away are they?
- Other infrastructure items in this particular joist bay (See the big footnote warning).
CAUTION: HERE BE DRAGONS!
There are all sorts of things that could be located between the ceiling of the first floor and the floor of the second floor. There may be wiring, plumbing and HVAC duct work. You need to use caution when running a saw through that space. Hitting hard plumbing or duct work is unlikely to cause an significant damage to it.
However, hitting wiring, soft plumbing (like PEX) or soft HVAC (like flexible, insulated duct work) can cause damage to what you just hit and could kill you.
If you hit a hard HVAC line, you'll either hit uninsulated metal and put a tiny scratch in it, or you'll hit an insulated duct and cut the insulation. Either shouldn't be more than an annoyance.
If you hit a flexible HVAC line, at a minimum you will tear into the insulation.
- An insulation tear within the heating/cooling envelope is more an annoyance, not a major issue. You may need to repair it, you may not.
- At worst, you'll cut all the way into the inner plastic line and you'll allow heated/cooled air out at this point. Again, more of an annoyance than a crisis, and something that's easily fixable.
If you hit a PEX water line you could cut into it and cause flooding. This, of course damages the drywall on the ceiling and the floor below causing more repair work.
- I'll be honest, I don't know the full implications of a cut in the outer surface of PEX. I don't know if it'll take that and not notice, or if you'll need to repair it right away because it could fail any minute now.
If you hit and cut into electrical cable, you could cause a short at a minimum and, at worst, you could electrocute yourself.
USE EXTREME CAUTION! Stop sawing at the slightest change in resistance or noise. Investigate what's caused this. Please don't kill yourself!