Endurance of Arduino UNO during 24/7 operations and mild power fluctuations
Is the Arduino Uno capable of surviving the extended service expectancy and execute the program correctly?

Yes, it is. Though your program and wiring needs to be stable and correct.

What is the influence of small voltage fluctuations on the endurance of electronics on the board?

Depends on what you call small. As long as you don't enter the brown-out region (where the voltage is so low, that the Arduino goes into reset for safety) or giving a too high voltage to it (See the specifications for that), you should be fine.

Is a heat sink needed for this application? How to determine the power draw to choose heat sink size?

The power draw of the Arduino depends on what you are doing with it. The best way would be to just measure the current of your application. Then you get the power by multiplying current and voltage. Normally you don't need a heatsink with an Arduino. The only part, that can get hot, is the linear voltage regulator. Depending on what you are interfacing with the Arduino, you might need to put a small heat sink on that (12V is a bit on the high side of input voltage for the regulator. It will dissipate the excess 7V as heat, so 7V multiplied by the used current). The microcontroller on the Arduino will not get hot (unless you are doing something terribly wrong).
And you should make sure, that the current for extra peripherals in your project doesn't flow through the Arduino, since that will place an extra burden on the voltage regulator. It should flow past it (means you connect them in parallel to the voltage source).

Is there any fatigue rating for the cyclic running of the program?

No, not really. I think the flash memory (where the program is stored) has a data retention failure rate of less then 1ppm (parts per million) over 20 years. So the program will not be lost, until you plan to run your application a hundret years. Also there is no read limit on the flash memory, only a write limit.