I will be installing both tile and carpet in my basement. Which should I install first? I'm primarily asking to determine what makes it easier in terms of transition between the two floors, i.e the doorway between the kitchen area and the den...

Install the tile first for the sake of the mess you could make on the carpet if done in the reverse order. Schluter makes an edge profile for tile to carpet called "Reno TK" which gets installed with the last tile at the doorway giving the tile edge protection and a spot to tuck the cut edge of the carpet.

Schluter-RENO-TK - Schluter-Systems (http://www.schluter.com/1_4_reno_tk.aspx)

If you use that, it needs to be done when you set the tile. Three other options include Z bars, the normal metal strips you often see... for those, you would place the strip on the slab, mark the nail locations, drill 1/4" holes into the slab, insert oak plugs and nail into that or, and I use this option pretty often...

Home centers sell oak seam binders with a bevel on both sides and predrilled countersink screw holes. I run it along my table saw to remove one side of the bevel, stain it and polyurethane it to match the cabinets, set it about 1/8" from the tile edge and mark the screw hole locations. Drill the hole in the slab and insert plastic sleeves and then screw the transition into the plastic sleeves. The sleeves are not anchors like you would normally see. It's more like a tubular fan folded plastic sleeve where the entire thing expands evenly around the hole, unlike regular anchors which expand outward in a "v" shape. The gap between the tile and the oak transition gets filled with a caulk that is color and texture matched to the gout. It also goes over and hides and protects from fraying, the cut edge or the carpet

Does this slab have any cracks, paint, sealers, old adhesive residue, does it absorb water?

The slab is in great shape, no cracks, no paint or sealers, it has not had any treatment, it does have expansion joints cut in it.

I did test the moisture content by taping a piece of plastic to the floor and leaving it for three days. No condensation at all on the plastic.