Ultrasonic Sensor using Uart Serial Communication?

SoftwareSerial mySerial(10, 10); // RX, TX

If you don't need RX or TX set the corresponding pin to -1 and not equal the pin of the other direction!

a complete 'message' should be 4 bytes long: a start signal 0xFF, a 8 bit signal, another 8 bit signal, and finally a stop signal 0xFF.

I read the datasheet differently:

The data of the sensor consists of boot code 0xFF (DATA1), distance signal (DATA2+DATA3) and Last Frame 0x00 (DATA4

The end byte is 0x00 and not 0xFF. So the output you get corresponds with the datasheet.

  1. Why is mySerial.available() > 0 required? The datasheet says the baud rate for signals is 9600, doesn't that mean no matter what comes out of the serial monitor, it will be in the format start_data_data_stop? It seems that here, the start bit is sent a bunch of times before a complete 'data transfer' is done.

The available() method tells you how much data already arrived (the number of complete bytes, not bits), so it's necessary to know if you already got data or if you still have to wait for them to arrive.
The rest of the question seems to be a wrong mixing of the terms "bit" and "byte".

  1. I'm moving my sensor around and the output doesn't change from what I copied from above. Is this likely a sensor issue, or is my code not parsing/missing/omitting some data being sent? Is this because I'm not using TX?

Moving around doesn't help if it already sees the minimal distance but it depends on how you move it around and what environment you're doing that in. The sensor is measuring a distance, not acceleration, so it won't detect movement (at least not directly).

I am still not able to get it to read an actual value when it is pressed against the bottom of a water bucket....

English is not my mother tongue so I might get the wrong sense from that sentence but from what I understand you're using the sensor wrongly. You have to mount the sensor to the cover of your water bucket (or a similar position if it hasn't a cover), the measurement direction downwards to the water surface.