Incorrect pronunciation of "privat" with current TTS (Jan 25 2015), it should be like this.
Appears to be working fine now! :) (Feb 17 2015)
Don't you ever pronounce the "t" on the end of "det"?
You can, and it varies a little by sociolect. Some people may be more inclined to pronounce it in a formal setting, or if speaking very slowly. These situations are rare, though, and in practice, the -t is almost always dropped.
As in, Q:"Vad är din ålder?"
A:"Det är privat!" ?
Sure, that works. :)