1) Fill the divot with either brick mortar or a combination of thinset (as a painted on bonding agent) and deck mud. Don't fill the joints.
2) How long was a "extended period of time" and how much moisture did you collect? The test is for 48 hours, and you should see nothing. A calcium chloride test can be performed to quantify the moisture. However, I believe that if you carpet isn't suffering from moisture issues, your tile won't either.
3) Schluter makes a strip that covers the end of the tile and carpet. that may work. A marble threshold will do, too. It's more a design thing than a technical one.
4) Depends on how deep and what you use to fill. You will find it easier to do floor prep as a separate step than tiling.
5) Once you get the slab prepped to accept thinset, any modified thinset will do.
6) I'm not overly concerned, either. Thanks for asking.
You asked about the water absorption: Your slab was probably sealed a long time ago, since water will go away, albiet slowly. Rent a floor scarifier or grinder and remove just the top most bit of concrete, and any high spots you have. You could also use a diamond cup wheel on an angle grinder, too.
You asked about filling the control joints in. You have to install your tile so there is a grout line over each of those lines. Those lines are caulked, not grouted. This allows the slab to crack at those lines if it wants to. If you feel like gambling with your money and labor, fill the lines with brick mortar and tile over them. I wouldn't do it, because I don't gamble willingly.
Let me ask this: have you considered acid staining this slab? Those control joints and some of the patched divots will give your floor a aged stone look that can be very dramatic.