I have used the exact same procedure that you describe on a number of occasions, and the result has always been a thick butter-like consistency, and I think that this is pretty much unavoidable when using a food processor. Maybe you can get a thinner consistency if you use the whole egg like ElendilTheTall suggested.
Anyway, since the sauce never emulsified, I think you have been pouring in your butter too quickly. It is important to just add a few drops at a time until the sauce has started emulsifying. After that you can go faster, but it should still only be a thin stream.
Also, something that I have noticed is that when using a food processor with only a small amount of egg yolks (or any ingredient, for that matter) is that the rotating knives will actually rotate above the ingredients and not actually mix them. When making hollandaise/mayonnaise, that will be fatal since you will just be pouring butter on top of a thin layer of egg yolks. So you might want to check your food processor and see if it works similar to mine.
While I realise that you use this recipe because it is easier than making hollandaise by hand in a bain-marie, I would suggest that you at least try making it "for real" once before settling on this method. The consistency is so much nicer. I think the flavour is better too but that may only be my imagination.
Temperature is the only foolproof way to determine doneness. Unfortunately it's not very practical to carry a thermometer over to your friend's house for brunch.
There are other indicators, of course, and you have mentioned the good ones for eggs.
I will add, however, that some indicators can be deceiving. For example, eggs releasing a pool of liquid is actually a sign of over-cooked eggs. The protein matrix is tightening to the point that it's actually expelling water. This is different than runny, uncooked egg.
Similarly, browning isn't necessary for the cooking of an egg (frankly, it's often detrimental). So while browned egg is likely more cooked than not-browned egg, it may also be less done on the inside.
They are clues you can use to identify doneness but none of them are guarantees.
I'd suggest sticking to fried eggs if that's what you enjoy and are more comfortable with.
So to answer your enumerated questions directly:
1) Scrambled eggs cooked to a typical doneness are moist, not-browned, and not expelling excessive liquids on to the plate. If the eggs are not done, they will not be "set" into curds and will be noticeably slimy.
That said, cook them how you like them.
2) Cooking time is dependent on cooking temperature. It's really not a good way to judge doneness. It works better for large cuts of meat or something that will take a lot of cooking time. An individual egg cooking time will vary. Yes, adding additional ingredients (mass) will slow it down.
Overall it will take just a couple of minutes tops. If you start with a hot pan, you can scramble an egg in just a few seconds.
Note: I'm talking about the scrambled eggs you typically see in American diners that look similar to the photos you posted. French style scrambled eggs are creamy in consistency and lack the large curds. Doneness is a little more difficult to judge in that dish.
Best Answer
I poach eggs, remove and chill them in an ice water bath to stop the cooking and then 'reheat' them in the simmering/boiling water right before getting ready to nap them with hollandaise and then serving them. (to aid in creating mass quantities of eggs benedict to be served at the same time)
I suspect that you can further cook your under-cooked hardboiled eggs with no adverse effects (even AFTER peeling) in the simmering/boiling water. Hope this helps.