Electrical – Can capacitors amplify signals

1: Yes, you can do that. Essentially, that's how power supplies work. They can handle multiple parallel networks within their current capacity. As for the heatsink, that depends on the regulator, the current draw, the ambient temperature, how efficient it is, etc. It's not a simple yes or no.

2: The capacitors depend on the regulator as well. Some require them all the time, some only require them depending on the input or output conditions, some never require them. The NTE1960 you linked to does not have an extensive datasheet, but is pretty similar to the LM7805. The capacitors are pretty much required for stable use. But these are linear regulators. Not efficient and they convert wasted energy into heat. Going from 12v to 5v, at say 700mA which is the high end for the RPI, that means 12 - 5 = 7v * 700mA = 4.9 Watts of energy being converted into heat. A heatsink would be required.

A Switching regulator is more efficient, in terms of both energy and heat. The OKI-78SR component you chose is a Switching Regulator. It shows that it would not need a heatsink in that same situation (Not in the engine compartment though, that's a different story). It is also a complete module, including the capacitors and the resistors it needs. It would be better.

3: A Car USB regulator would work just fine for your case, as long as the draw on it is under it's maximum. Some are 500mA, some are 1A, or better or in between, but some can't actually supply the amount of current it says it should, so you would need to test. The Model B has a 700mA draw/limit, the Model A is 500mA. Most of these usb regulators are switching supplies, and for your purposes, a car usb adaptor would be exactly like the OKI-78SR. At 4 bucks for the OKI-78SR (plus shipping) compared to a few bucks for a car USB adaptor, it really just depends on which you can get easier. Even retail, you can get a decent car one at any convenience or auto store for 10 bucks.

You could even gut the car USB adaptor for the board inside. Those things are so small now they are smaller than a car cigarette lighter, with the case, and the size of an SD card without the case.

Adding to @GeorgeHerold's answer:

1. You need to include frequencies down to below 30Hz. Most of the energy in a typical heartbeat is between 30 and 40Hz. Given the cutoff for the highpass that @GeorgHerold computed, you are losing a good 15dB at the lower end.
2. Try using a directional microphone instead of an omnidirectional. Directional microphones have two openings, one in front and one in back by the solder pads. I find I get much better results by installing a directional mic such that the front (where you normally speak into the mic) is inside the tube, with the port on the back of the microphone out in open air.

Actually, I don't use a stethoscope in my experiments. A simple plastic shell with a hole in it for the microphone works very well. What I've used for this is a plastic cap like you would use to close one end of a plastic water line. I had some laying around that are about 2" in diameter, and they work well.

I find that a directional electret microphone installed in a platic cap as described works well enough that I can capture heart sounds using nothing more than a PC or smartphone with no additional electronics - I can then do filtering, recording, or other processing digitally.
The PC soundcard has a 20dB boost on the mic input, but the smartphones don't. Since the cards use 16Bit sampling, you have plenty of headroom to digitally amplify the signal even without using the boost. If you resample from 44100 down to 400 or so, then you could probably gain the equivalent of a few bits