It does sound like there might be a problem with the concrete portion of the room but we would also like to know as much detail as possible on how hvac is laid out. I hate chasing rabbits all over then be told one was hidden beneat a bucket from the start. It appears you arn't looking to jump right on this so I'll only get you started with seeing what can be done and how much effect it will have. The nuts and bolts of remediation can come later,depending how much your enthusiasm wanes after stareing down the barrels of realality. Use quick reading digital thermometer to read A outdoor ambent, B air temp in inclosed space beneath floor C soil temp beneath floor at 2" 6" and 12" below grade (test near center of room and midways between columns,not next to columns nor block walls.) Repeat soil test within 2 inches of 2 or 3 columns near center of room and 2 or 3 locations 2 inches from inside of block wall. Record and average readings for each test set. You can't use a pos device nor curupt readings with breath,body heat,lights and such. The differentials can be much smaller than you are acustomed to in daily life but are meaningful non the less. Try doing this after 3 or more days ( longer the better) of fairly consistant sun.wind,precipt and temps ( day night flux is inevetable as long as there isn't changing weather patterns.) A good thermometer isn't expensive and can be used for many different things for years to come. I try to keep secret the device I used for steak last yesterday was used last week on a boiler system. After you assemble and chart your data, you should have an aah haa moment.

The driver on basement floor is often the loss of headspace from putting insulation on the floor. Various type of insulation boards are often used since they have the highest R-value per inch allowing for smaller thicknesses. Wood sleepers are then placed about every 16 inches and a layer of plywood applied.

Be sure you understand the options. There are 2 kinds of crawlspaces allowed by the IRC: Vented and Unvented (aka "sealed"). The terms "unvented" and "sealed" are misleading because this type of crawlspace is not completely sealed or unventilated. The IRC code requires that both of these types of crawlspaces be "ventilated" so that moisture does not collect and encourage the growth of fungus (whether from the ground, condensation on the slab, etc.) by one of the following methods: 1) "Vented" Crawlspace - install vent openings in the exterior wall of the crawlspace 2) "Unvented" Crawlspace - inside the house supply to or exhaust from the crawlspace a small amount of air with a return air path to or from the house and insulate the exterior walls of the crawlspace. Both crawlspaces must have a vapor retarder at ground level and an access opening. For more information see IRC section R408.1, 2, & 3.

If you want tile, go for it. And yes, tile is cold. Yes, concrete is cold. If you live in a HOT climate (like Nevada) then you should be fine. If you live in Seattle or NY City, then this is going to get old (and cold) very quickly. A VERY large area rug (ahem...rugs everywhere) will reduce the "cold" sensation. But not eliminate it completely. It all comes down to your location...just like real estate.