DIY ultrasonic sensor questions
The transmitter (what you are showing) is the easy part. The part you have circled in red is the transducer. It is a piezoelectric disk, typically in a tuned cavity that resonates at some particular frequency such as 40kHz. Here is a datasheet of one such device (optimized for 40kHz). Electrically, it represents a capacitive load, plus it generates some voltage by itself once it starts moving. You can try simulating it with a capacitor of a few nF and see if the driver at least gets to +/-V close to the supply voltage within the half period of the transmitter waveform.
Rather than use that driver circuit, I suggest using a MOSFET gate driver chip such as the MCP14A0155.
The mechanical Q of the transmitter and receiver transducers have a great deal of effect on the behavior if you're intending on sending a sharp beep out and hoping to receive something relatively sharp. It will also be affected by the drive circuit. The transmitter datasheet, such as it is, is a bit sketchy on such matters.