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The timer in your link doesn't start the time with the sound of a shot. I've used that model in club matches many times.
The time starts with the beep when the start button is pressed. The timer "hears" each shot and records it to memory. There is no stop button unless a switch is used on a "stop plate" target. When the shooter stops shooting, the timer has already recorded all the shots and calculated a total time for the course of fire regardless of the number of shots fired. You just look at the timer to get a total time.
70 feet is probably too far away for a timer to reliably register shots. Best is for a partner to start the shooter from a safe position behind the shooter and follow him through the course. When I practice alone, I use my timers pocket clip to clip it to my pocket and shoot as normal using the random delay. I'll review my time after the course. Works just fine.
Hit factor is easily calculated. Lets say you have a course that has a maximum of 100 points. Lets also say you shot this course with a score of 95. This is 5 points down. Lets say you shot this course in 15.00 seconds. You take your points (95) and divide it by your time (15.00) to get your hit factor. 95 divided by 15.00 is a hit factor of 6.333. Most timers are capable of calculating this as you described.
Stage points come into play when scoring matches. The shooter who has the highest hit factor on a stage is awarded all the points available on that stage, in this case 100. All other shooters are awarded stage points as a percentage thereof according to their hit factor, with the above shooter being 100%. The shooter with the most stage points accumulated over all the stages wins the match.
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