Need Assistance Wiring a Duct Booster Fan
Should I just pay them the $700 and be done with it?


Depends on ..


  Am I too far out of my depth?


Can you operate a screwdriver, watch you tube, turn off a circuit breaker .


  Am I likely to be bumping up against so many safety and code issues
  that I should just pay to have it done?


Only the fan matters - UL Listed, GFCI Circuit if the location has high humidity like a shower (probably already is if that is the case) 


  Many of the in-line duct fans I've seen plug directly into a 120v
  outlet. I was thinking I could just rig the relay to energize a 120v
  outlet and plug the fan into that. In this way, I can just plug a new
  fan in if/when needed. I know codes differ from location to location.
  But is controlling a 120v outlet with thermostats and a relay ever OK?


Why do this you don't need to! 
Your current fan is 120V or possibly but not likely 220v - just use the three wires on your current fan (Black , White , and Green/copper) which are probably wire nutted in a junction box by the fan motor and swap out with a new fan.

You can probably buy a new fan of the exact same type as yours online for $100.. Do an online search of the model number found on the motor faceplate. 


  Let's say #2 is a no-go. I need to identify a proper relay. Can I use any old 24v to 120v relay to power the fan?


Your HVAC relay that is 24V - that is the solenoid side - not the load side - the load side is 120V or 220V and is what turns on your fan.

I'd love for all of this to be stuffed into a junction box, but it looks like any relay I see is mounted outside of the junction box. I assume this is to keep the low voltage and line voltage separate. So I take it that it's never OK to hide the relay inside a junction box?


  You misunderstand: the relay is inside the junction box with Low
  voltage on the COIL side (which pulls in the contact side) and 120VAC or 220VAC on the Contact side.
  This is perfectly fine and good that they did this.


Here is an electrical Diagram of what you probably have and should see..