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.
Best Answer
You have to check the datasheet for the specific regulator. Some require a certain level of resistance in series with the capacitor, and recommend electrolytic (or tantalum) capacitors due to the high ESR. However, ESR is specified as a maximum, and future capacitors with far lower ESR may still be within their own specs but have too low ESR for the loop stability.
These types can be used with ceramic caps, but to ensure stability, it is necessary to put a small series resistance (typically <1ohm) in to ensure stability. Other LDOs are specifically designed to be stable with ceramic caps.
Note when using ceramic caps that you won't always get the capacitance you're paying for. Using class 2 ceramics for supply bypass is worst case in that your AC voltage across the cap is minimal (lowering the capacitance) while the DC voltage is probably close to the rating (further lowering the capacitance). This is a good tool to see those effects...ceramic chip caps will behave about the same for the same size and value from different vendors.
Although as a general rule more capacitance means better performance, each regulator has its own idiosyncracies. Also remember that the output capacitance doesn't just include the capacitor closest on the schematic...it includes all the bypass caps connected to that supply.