I've heard that mobile phones send signals to the tower even when the mobile goes off so your location can be detected. Is this possible?
These articles state it can be done:
On most Nokia phones, the alarm still rings even when the phone is turned "off."
It can, if someone with the needed technology, like the FBI, wants it to and installs the needed device or software without your knowledge. This can be avoided by cutting off the power supply to the device, i.e. remove the battery from the device.
Cell phones can be taped and used as a listening device as long as their buttery is in, even if they are powered off. As this article by cnet demonstrates:
Countermeasures against it are taken in the IDF (Israeli Army). I have experienced in first hand that to classes, briefings or meetings where confidential material is talked about or revealed, all participants must suppurate their phone from its buttery, and if your phone doesn't allow this, e.g., the Iphone, then you leave it outside of the room. Here is an article (in Hebrew) talking about information safety in the IDF that tells the same thing:
Which translates to:
*: "Mountain Rose" is a secure IDF cellular network.