Ultrasound in air ranging system design
The goal is to get ~1mm accuracy


Wavelength is determined by speed and frequency. Speed is approximately 340 m/s and therefore wavelength is 8.5mm.

So what you may ask. Any standing waves you might get will occur every 8.5mm and these could ruin you expected accuracy of 1mm. 

You may then point out that you will use a pulse driven into the transducer. The 40kHz resonators I've come across are very "resonant" and generating a pulse may not be that easy.

I'm saying these things because I think you need to take them into account.

A narrower beam seems logical to me or else there could be several reflections from different objects coming back and obscuring your desired distance measurement. Also remember that narrow beam devices can still produce/be susceptible to side lobe interference.

As for your other questions I think you need to determine what you want to transmit before you think about signal processing.