Reverse engineering and reimplementing a garden hose water timer
Yes, it does. It's a simple oscillator that can be turned on and off by the microcontroller. An oscillator that switches Q3 on and off is obviously required for the inductor-diode-capacitor circuit to function as a boost converter (you can read plenty about their operation anywhere), the only peculiarity of this circuit is that the oscillator is built from transistors and an RC-network. Alternatively (and nowadays usually) some kind of special boost converter IC could have been used, but I guess the ICs are way more expensive than a bunch of simple components, especially when you buy simple components in large quantities.




For better understanding of the circuit I redrew it (some component values are changed for better simulation, so all of the results are for qualitative study only):


And simulated it to confirm the assumptions about its operation (open in full size to avoid scaling issues):


The first plot (you can also see all the probes on the schematic):


Red trace = microcontroller signal (ON/OFF the booster). I switched it high and low several times just for example, the real uC just switches it for a constant amount of time (known to charge the cap to the desired voltage) high.
Green trace = the base of Q2, which effectively switches Q3 (this arrangement is known as the Sziklai pair).
Blue trace = capacitor voltage.


Second plot: capacitor voltage build up when being charged with this circuit for a long time (1 second).




Yes.
Yes, this is probably a protection circuit placed there in case the solenoid loses contact (therefore the cap won't be discharged) and the circuit tries to charge the capacitor over and over (up to voltages dangerous for other components). I am not sure why there is a diode. May be they don't want the capacitor to discharge below ~0.7V? But why... Actually, it would make more sense if it was a Zener diode placed the other way around. Have you rechecked its polarity?


What do you mean "a pin diode"? The PIN-diode, or the package? In case of the latter, it is not a through-hole diode, it's a surface-mount one, just like the others, but in a different (though common) packaging. Why? Well, there could be various underlying reasons, that are not necessarily important for the schematic itself (the cheapest one they could find meeting their requirements just happened to be in mini-MELF). Anyway, it's necessary to understand its function, before trying to answer this question.

Addenum: If I were you, I would leave all the existing schematic in place and just add wireless control on top of it. Why bother reinventing the wheel?



Edit: To clarify things. First, initially I didn't pay attention to component designators while drawing the schematic and it created some ambiguity in transistor notation. Fixed now. Second, some simulation engines require (especially when some switching stuff is going on) equivalent circuits of real components to be explicitly drawn. In Proteus an inductor primitive has no resistance and so does a capacitor primitive, however, in reality both of them have equivalent series resistance (ESR). Sometimes during simulation this lack of ESR in ideal component models can cause infinite currents and other nonsense to appear and crash the simulation. That's why I added some dummy resistors. Now I've labeled them accordingly too.