Be aware that with switches and 3ways white is not necessarily neutral. White can be used as a traveler or as a switch leg.

ugggh. foosball, allow me to skip all this and answer your original question! YES. If you have power in that box that "continues on to an outlet", that means you've got a hot feed and a neutral in there, which are what you need. To achieve this: Inside the switch box, remove the ALWAYS HOT wire from the already-working switch. Add a wirenut and two pigtails - one going to the existing switch, the other going to the COMMON (black) screw on the 3-way switch. There should already be a wirenut or other splice attaching the incoming neutral and the outlet neutral. Add the white wire from your 12/3 to that splice. Connect the black and red wires of the 12/3 to the remaining terminals of the 3-way switch. Inside the ceiling fixture where you've run the 12/3 to: Connect your fixture's white wire to the white wire of the 12/3. There must be another piece of 12/3 that goes from the fixture to the remote 3-way switch. This cable's white wire should be permanently relabeled hot (use a black sharpie) and connected to the fixture's black wire. At the remote 3-way, relabel the white wire and attach it to the COMMON terminal. Attach the black and red wires to the remaining terminals.

They exist. More common to find ones that can be used as 3-ways than those who can't, however, you'll probably have to replace both of the existing 3-way switches with the matching master/slave for the wireless switch.

Non-contact tester can give false positives under the right circumstances, which, I guess, is better than false negatives. ;-) Digital multimeters will also detect "phantom" voltages but those usually show up with readings in the 50-70 volt range, which is a clue that something is amiss. Like any tool, each has it's advantages and disadvantages. If you're going to do a lot of electrical work, it might be useful to invest in a solenoid type tester. Your local home center will have them for about $30.