The 2nd op-amp (feeding the BJT) isn't a comparator - it's a voltage follower - whatever voltage is on the +Vin input gets seen on the emitter of the BJT - notice the negative feedback from the emitter - it's a gain of unity buffer amp.
The first op-amp is a camparator as your question states.
Page 1 of the data sheet says it all. \$V_{OUT} = F_{IN}\times R_1\times C_1\$ where \$R_1\$ and \$C_1\$ equate to your R46 and C40 on your 2nd drawing. This is the capacitor the charge pump is charging.
C42 does two things - the higher the value the lower the ripple voltage will be seen at the output however, the longer it will take to attain the correct voltage should the frequency change. Read towards the bottom of page 8 on the data sheet.
I think any engineering design is handled top-down when you really look at the design process. This is not limited to digital electronics or even electronics at all.
Design of analog circuits is no different. You start out with what a circuit needs to do, then you come up with a overall strategy to solve the problem, then keep on drilling down until you get to whatever low level details your kind of design requires. For every engineering design I can think of, there are additional levels of abstraction both above and below the design. Again, analog electronics design is not different in that regard than any other engineering discipline. The same general process and existance of multiple levels of abstraction applies whether you are designing a audio amplifier, digital computer interface, the front wheel suspension of a car, or the cooling system of a nuclear power plant.
Let's use the design of a audio amplifier as a example. Higher levels above the design come to you as specifications. There is some limit above which you don't get to make choices (perform engineering design). The need may be to drive a small speaker so that someone within proximity can hear voice, like something built into a automated teller machine at a bank, or a gas pump. If you work for the ATM company, then you don't get to decide you want this to be a 300 W HiFi audio system for a theater.
Often determining the true requirements is part of doing a design. The company has decided to put a speaker in the new ATM so that it can issue voice prompts. They haven't said how many watts must go into what kind of speaker. This is your job to figure out. On the other hand, where the voice signal comes from may have already been determined and you don't get any wiggle room. All you can do is make sure that interface is clearly documented.
Once you have the requirements well decided and documented, you sit back and think of various high level ways to solve the problem. Pretty quickly, you'll narrow this down to a few alternatives, like getting a off the shelf class-D audio chip, a whole module that pretty much does everything between the source signal and the speaker, or something you design from lower level building blocks.
Good engineers don't just take the requirements at face value if they see something that might possibly be a small change externally but that could be a bigger advantage to their design. For example, you might prefer the audio came at you as digital words over SPI or something, but it is specified as line level analog. You go to the engineer that designed the circuit that produces the line level output. Perhaps you find that he actually has digital audio and is running that into a D/A because he figured that's whatever is downstream would want. Feeding digital audio to you not only saves money in his design, but also in yours.
This sort of looking at the bigger picture one or two levels up and trying to do the best for the overall system is all too rare nowadays. This is something the chief engineer of the project should be looking into too, but chief engineers are rare nowadays too. Often you get a project manager with little engineering skill, with nobody really coordinating the engineering effort between disciplines or subsystems.
Anyway, after you look at a few alternatives, you pick one going forwards. Let's say you decided that the does-all module is too expensive for the volume of this product. Your own discrete solution would work, but take more board space than it should, especially since it would be inefficient power-wise and dealing with the heat in the cramped space you have would be a real problem. So you go with the class D audio chip, except that none of the ones for the required power level can work directly with the power supplies you have available.
Now you have decomposed the overall amplifier into some power supply conversion, and the class D chip. Each of those will have details, etc. Eventually you get to a schematic of parts you can buy and put on a circuit board.
That may be the end of your design, but of course there are many levels below that. You're going to lay out the board, but you're not going to design the details of the physical board yourself and the process that will be used to make it. You specify holes and traces, and someone else makes sure that milling, drilling, plating, etc, all happen according to your spec. You buy a class D amplifier chip, but there is obviously significant design inside that block. If this is a super high volume product, then you might design a chip with a class D amplifier being just a part of it. Even then, someone else will design the silicon fabrication process so that you can simply talk in terms of transistors instead of masks and interconnects and doping levels and the like.
I don't see anything different here just because the design happens to be a analog circuit.
Best Answer
I'll start with the 555.
The circuit around the 555 is pretty much just the classic "astable multivibrator" circuit as given in the 555 datasheet:
\$R_A + R_B\$ in combination with \$C\$ sets the frequency.
If you change \$R_A\$, then you change the frequency of the oscillator.
\$R_A\$ sets the current used to charge \$C\$. More current (lower \$R_A\$) means a faster charge time which means a higher frequency.
Now, look at your circuit.
\$R_A\$ is missing. Instead there's one of the transistors where \$R_A\$ belongs.
Those two transistors form what is known as a current mirror.
The current through the right hand transistor (in place of \$R_A\$) is the same as through the left hand transistor.
The current through the left hand transistor is governed by the 39k resistor and the voltage at \$V_{in}\$.
The two transistors are actually two (danged near) identical transistors on one subtrate in one housing. That's the "CA3096AE" marking.
Two separate transistors won't work as well for the current mirror because they aren't as close to identical. The current mirror depends on both transistors having the same exact characteristics and on both being at the same temperature.