(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?

No, the evidence that the keys were found is strong enough to outweigh the problems in the searching method.

(b) Should these considerations affect your subsequent decisions (e.g. to go out, locking the door behind you)?

No.

(a) Once the lottery results are out, do you check your ticket? Why, or why not?

Personally, I'd probably check it for fun. It would take less than a minute, and provides enough amusement to be worth the time. I could easily see myself not checking if I forgot, though.

(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?

Very confident, let's say >99.9%.

No other hypotheses are raised to my attention. You could ask for hypotheses conditioning on me not really winning, but none of those are likely enough to outweigh the 1/100 million chance of actually winning.

Perhaps I'd check the numbers a few other times, using proxies to rule out the "hacked" hypotheses, but that's only because that's easy to check, not because I would take it seriously.

(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?

The point I become confident is after I check the numbers.

Edit: after reading devas's comment: I had assumed that the lottery ticket was state-run and well-known. If not, I would be a lot less confident (in fact, I've already won dozens of these lotteries by virtue of having an email account! Aren't those people who randomly choose email accounts to win money so nice?)