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.
Baked, covered, at low heat :
- ovens self-regulate, so you're going to get a more consistent result each time than doing it on the stovetop.
- covering will minimize evaporation, which will cool the top more.
- low heat will minimize problems with the edges cooking before the middle ... but eggs also have this strange thing where it's more difficult to over cook them at low heat ... and it's not just an issue of carry-over cooking (which lower heat will help to minimize, too).
As for shape of the vessel, I'd go with a rectangular baking dish -- because then I could more easily portion out 16 equal sized servings than from a round or oblong vessel.
Best Answer
Those 4 vegetables (and fungi) all require different cooking times. Chop the onion and pepper into dices of roughly the same size, to ensure even cooking. Get them frying, with a little salt which will help to soften them, over a medium heat while you slice the mushrooms, then add those: it is more or less impossible to overcook a mushroom incidentally.
Once the onion has softened but still has some bite, chop the tomato and add that, then add your eggs more or less straight away - the tomato doesn't need much cooking.