You don't need to run a air temp sensor. But if you have one, your tuner may choose to use it, if he has time, to do some fine tuning. Water temp sensor is needed.
With the Toyota ECU, you use the standard 20v air sensor, and the 4A-FE one is the same or similar enough to use also. The 16v one looks very different, but could probably be used.
With a Adaptronic, you can use any sensor, and calibrate it in the software, and the stock 20v one is often used.
With Haltech, I can well imagine that a few years ago they required to you use a generic GM or Ford sensor. I would hope their current ECUs allow for calibration to the sensor you have.