Can Our Cell Phones Track Us?

"Can a government agency use cell phones as eavesdropping bugs? ...if the phone is on or off?"

Generally speaking, if the cell phone is truly off (no power applied), then it is useless as a surveillance device... unless it has been specially modified.

If it is "on," but not being used in an active phone call (in standby mode) then it can be used to track the phone's location in many service areas.

I am not aware of remote turn-on of microphones (as a normal feature) in any except GSM type cell phones. If this is a concern there are two solutions: Keep the phone turned off and/or remove the battery when not in use, or keep your phone into a special radio-frequency shielded pouch so it can't communicate.

This is not to say cell phones have not been used as bugs. Modified spy phones are being sold on the Internet. The look like regular cell phones. They work like regular cell phones. But... they allow an eavesdropper to call the phone (silently) and listen into on-going calls and anyone speaking near the phone - even when it is not in use!