1 (a and b) Not to speak of. The key difference is that the event of finding your keys is quite different from that of (say) getting a significant result in a medical trial. When I have what looks to me exactly like my key ring in my hand, with all the usual things on it, the possible failure modes (hallucination, someone having maliciously planted a near-duplicate in my house, ...) are both rarer and weirder than when I have just found that 240 of the 400 patients in the treatment group got better, versus only 215 of the 400 in the control group. In the latter case there's a 1% chance (or whatever) of getting the result merely by chance, and if the experiment is poorly designed in other ways there are plenty of other demonstrably-high-probability ways to get misleadingly good-looking results.
All those methodological "errors" might reduce the odds that I've really found my keys by, let's generously say, a factor of 100. But once I have (as it seems to me) the keys in my hand, unless I'm knowingly impaired by drink or drugs or something, that's probably at least 1000000:1 evidence that I've found them. So even a 100:1 discount, which would kill most published experimental findings, would leave me something like 99.99% confident of actually having my keys.
Other relevant differences, more relevant to (b) than to (a) and perhaps also relevant to whatever point you're trying to make: once I've got my keys I will then try to use them and thereby rapidly get more evidence for (or, improbably, against) the correctness of my experimental result; the consequences of an error are rather minor compared with (e.g.) those of thinking a treatment cures cancer when it really doesn't; bringing in some impartial other person to replicate my search is a bigger increment of relative effort than arranging a replication of a typical scientific experiment.
2 (a) Yes. A typical prize with that sort of probability might be on the order of $10M, for an expected benefit of 10c from checking the ticket. It takes only a few seconds to check. (b) First I would double-check it, because the single highest-probability way to be wrong is self-deception. There are other hypotheses but empirically they seem to be improbable: e.g., I would expect a non-negligible fraction of cases in which someone is convincingly hoaxed into thinking that they have a majorly-winning lottery ticket to get publicized (because it seems like it makes quite a good story). So I might be, I dunno, 80% confident of having actually won at this point, which is plenty enough to justify attempting to claim. Note that the relevant probability to compare against is not "someone carried out a successful lottery hoax" (or whatever) but "someone carried out a successful lottery hoax on this occasion with me as target", which is much lower in the same way as the probability of "I just won the lottery" is lower than that of "someone won a lottery once". (c) I think I would define "really did win the lottery" so that once I've had my claim checked and been paid there's no further question (barring extreme options like having hallucinated the whole thing). Until that point I would still entertain the possibility that some kind of hoax or error is at work, though after getting confirmation from the organizers I don't think this would make a big difference to my actions beyond encouraging me to check the subsequent steps in the process.
( I know you said you want numerical answers to everything, with good justification for them all, but you haven't provided any reason why I should actually put in the effort to do that and I have chosen not to.)
Answers to these questions should be expressed numerically, where possible, but no number should be given without a justification for the specific value.
1. Suppose that you have mislaid your house keys, something most people have experienced at one time or another. You look in various places for them: where you remember having them last, places you've been recently, places they should be, places they shouldn't be, places they couldn't be, places you've looked already, and so on. Eventually, you find them and stop looking.
Every time you looked somewhere, you were testing a hypothesis about their location. You may have looked in a hundred places before finding them.
As a piece of scientific research to answer the question "where are my keys?", this procedure has massive methodological flaws. You tested a hundred hypotheses before finding one that the data supported, ignoring every failed hypothesis. You really wanted each of these hypotheses in turn to be true, and made no attempt to avoid bias. You stopped collecting data the moment a hypothesis was confirmed. When you were running out of ideas to test, you frantically thought up some more. You repeated some failed experiments in the hope of getting a different result. Multiple hypotheses, file drawer effect, motivated cognition, motivated stopping, researcher degrees of freedom, remining of old data: there is hardly a methodological sin you have not committed.
(a) Should these considerations modify your confidence or anyone else's that you have in fact found your keys? If not, why not, and if so, what correction is required?
(b) Should these considerations affect your subsequent decisions (e.g. to go out, locking the door behind you)?
2. You have a lottery ticket. (Of course, you are far too sensible to ever buy such a thing, but nevertheless suppose that you have one. Maybe it was an unexpected free gift with your groceries.) The lottery is to be drawn later that day, the results available from a web site whose brief URL is printed on the ticket. You calculate a chance of about 1 in 100 million of a prize worth getting excited about.
(a) Once the lottery results are out, do you check your ticket? Why, or why not?
(b) Suppose that you do, and it appears that you have won a very large sum of money. But you remember that the prior chance of this happening was 1 in 100 million. How confident are you at this point that you have won? What alternative hypotheses are also raised to your attention by the experience of observing the coincidence of the numbers on your ticket and the numbers on the lottery web site?
(c) Suppose that you go through the steps of contacting the lottery organisers to make a claim, having them verify the ticket, collecting the prize, seeing your own bank confirm the deposit, and using the money in whatever way you think best. At what point, if any, do you become confident that you really did win the lottery? If never, what alternative hypotheses are you still seriously entertaining, to the extent of acting differently on account of them?