The link you gave includes a link to the actual federal regulations (14 CFR 250) which clarify some of these points.
As you saw in the link you gave, the compensation is either nothing (if delay less than 1 hour), 2x your fare with a cap of $775 (for delay of 1-2 hours), or 4x your fare with a cap of $1550 (delay of 2 hours or more).
Does the delay include ANY flight United gets me on (including those with layovers in for example, SFO, EWR, LAX?) or does it only apply to flights of the same stop amount (since this is a direct flight, the delay would be calculated based on the next nonstop flight United can get me on). Please leave comments if this makes no sense.
"Alternate transportation" does not have to be the same number of stops. The regulation assumes that you only care about when you arrive at your final destination, not how you arrive there. And it's based on the transportation that the airline offers, not on what you actually end up taking.
So if for instance, you originally booked a nonstop flight IAD-LAS arriving at 9:00, and United offered to rebook you on IAD-SFO-LAS which would arrive at LAS at 10:30, you would be eligible for 2x compensation. If you say "I hate connecting flights, so I'd rather take the next IAD-LAS nonstop, and the next one would arrive at 12:30", the airline will probably oblige you, but your compensation will still be based on a delay of 1.5 hours, not 3.5 hours.
How does United calculate the one-way fare for compensation? Is it based on the price at the time of booking or what?
It's the fare that you actually paid. If you bought this flight as part of a round-trip or multi-city itinerary, you should be able to get a detailed receipt from the airline that shows the breakdown into the one-way fare for each leg, and for each person traveling.
Also, my family of three is travelling. Would we receive compensation individually (three compensations) or all together (one compensation)?
Compensation is per person. So in principle, they could bump just one of you. Then that person would receive compensation based on the amount of the fare paid for that person alone (which again should be broken down on your detailed receipt), and the other two would be able to fly as normally scheduled.
But splitting up families often leads to stories that generate bad PR, so I expect they try to avoid it. What's more likely is that, if they just need one seat, they would bump someone else who is traveling alone. They're allowed to take such factors into consideration - it doesn't have to be a totally random lottery of all the passengers. If they need three or more seats, then they might bump all of you together, and then each of you would receive compensation based on your own fare, thus effectively getting 2x or 4x the total amount paid for your family.