Strange minimalist power supply, which IC?
&quot;L1&quot; is not an inductor, it's the world's cheapest looking magnetostrictive ultrasonic transducer.  The construction is similar to a solenoid or other speaker, but there is a rod of magnetostrictive material in the center (which changes shape slightly when a magnetic field is applied, in this case lengthening and contracting).  This rod is capped with a smooth, non-magnetic metallic piston which acts as the surface that actually vibrates the air.  The piston can be seen clearly in your photos.  If it were an inductor, it would be made of ferrite and the top and bottom caps would be ferrite as well, not just a plastic bobbin as it appears to be in the photo.
It is claimed that certain frequencies (like 38kHz I believe) are thought to repel mosquitos and other things.  Unfortunately, this appears to be something that someone just made up one day.  There is no evidence that this claim is true, or that any frequency of sound repels pests like mosquitos, gophers, or whatever claim a given device is making.  The most these things do is annoy your cat or dog to high heavens probably.
In the US, the Federal Trade Commission has investigated several sonic repellent makers for false advertising, so the claims seem very suspect.
There is no interesting or novel power supply stuff going on here, unfortunately.  Most of the circuit is simply deriving a 3.3V DC power supply for the IC in the most horrifyingly inexpensive way possible.  I don't know why you think there is an inductor in parallel with capacitor C1, it sure looks like a 400K resistor to me.  I'm guessing the capacitor is maybe 0.47µF or in that ballpark.  R5 is only there to limit the inrush current into C1.  Due to the impedance of the capacitor, the current though them (remembering that capacitors will conduct AC so they act like DC-blocking resistors that don't dissipate the majority of the current as heat like a true resistor would.  This is nothing more than the series resistor in your standard zener diode power supply, albeit with the twist that there is a half-bridge rectifier in between to convert it to DC.
The current is limited to \$ I = \frac{V}{\sqrt{R^{2} + (\frac{1}{2\pi fC})^{2}}} \$ where \$ R \$ is the series resistance (R5), \$ f \$ is the frequency, \$ C \$ is the capacitance in Farads, and \$ V \$ is the RMS voltage.  Depending on the value of C1 and if this is designed for 120VAC or 240VAC this would limit the current in the ball park of 20-50mA.  This lets the zener finish the job and regulate the voltage to 3.3V (with a few hundred millivolts of ripple - cheap and gross).
The switch turns the blue LED on and off.  That is all it does.  The reasons why are shrouded in mystery to me.  I guess if the blue light is too bright/annoying (and considering it is a blue LED, it probably is too bright and annoying), that lets you turn it off.  Or maybe it is a placebo button that just gives the user the impression of turning something on and off without actually doing anything.  Who knows.
You do have your transistor drawn incorrectly, pin 5 of the IC is almost certainly going to the base of the transistor.  This is the power element that actuates the transducer.  The IC itself is not an IC at all, and it is possible it never had a label to begin with rather than having one sanded off.
The pinout matches one of the standard arrangements for a 4 pin oscillator - pin 1 is NC or occasionally an enable pin, pin 4 is VCC, pin 8 is GND, and pin 5 is clock out.  Pins 2,3,6 and 7 are likely not internally connected to anything, which is why they are left floating.
It would normally would be more recognizable due to the metal can packaging, but this is a product where every expense is spared, and they probably had the part custom packaged in a DIP because that was the cheapest package compatible with a through-hole assembly process.  On a real oscillator, it would instead be pins 1,2,3, and 4 and only have 4 pins.
It outputs a buffered logic level output at some frequency in the ultrasonic range which drives the transistor and emits an ultrasonic tone (at least when the 60hz AC waveform permits).  This ultrasonic frequency promptly does nothing at all to any mosquito, but if you wanted something to anger your cat while burning maybe a watt of power whether or not the switch is 'on' or not, then this design is a highly cost-optimized way to do it.
I'd desolder the transducer and see if I could do something interesting with it and just toss the rest.  The DIP oscillator isn't anything worth your time.