What is the earliest a team can clinch a division?

I'm going to focus on the first question; if you want to ask more than one, you should ask separate questions. I'll touch on the others briefly at the end, but they're much more complex.

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The earliest realistic time a team could clinch a division (meaning, be ensured of first place) would be with 9 wins and every other team having 2 or fewer wins (so, 9-0 and no other team better than 2-7). That's because of how scheduling works - playing twice against each other in the division plus the desire to not have the same teams playing each other in nearly consecutive weeks means that teams nearly always play each divisional opponent once in the early weeks.

If Team A goes 8-0, and three other teams, B,C,D, each lose to every other team outside of the division, and split the games equally(so B>C, C>D, and D>B), then you'd have a scenario with 8-0 / 1-7 1-7 1-7. But that would not be clinching - you'd still have the possibility of losing out to be 8-8, and one other team winning out to be 9-7. (Theoretically if no other team in your division had an intradivisional game, you would be able to be 8-0 with all other teams 0-8, but that doesn't happen in general from a scheduling point of view.)

Thus, you need to be 9-0 with no other team better than 2-7; at that point you would be no worse than 9-7, and no other team would be better than 9-7. You'd still have to set things up schedule-wise to ensure you had won the tiebreakers over all of the 2-7 teams, but that's technically possible (and, in fact, all the other teams could be in theory 1-8).

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What's the earliest week you could clinch a wildcard or a bye is not really practical to say; there's too much complexity. It would be later than week 9, in any event, as 9-7 is not always a wild card, and there's too much complexity to have everything break properly. Most likely a 10-0 team will have effectively clinched a wild card, though very occasionally 10 win teams don't make the playoffs (The 2008 Patriots at 11-5 actually missed it, in fact.)

Clinching a bye similarly has too many complications to give a realistic answer to; the second best team in the conference practically speaking has always had 10 wins, I believe. Technically it's possible to get in at 9-7, or at least 9-6-1 (see the 1989 Browns, though that's before the current structure and schedule), but it seems improbable. Usually the #2 bye is a 11-5 or better team. As one example, the 2007 Patriots didn't even clinch their division (an extremely weak AFC East) until their 11th win (11-0). With a 13-3 team, an 11-5 team, and a 10-6 team leading the other three divisions, they likely didn't clinch a bye until their 12th win of the season; that seems right in general.

And as far as clinching a playoff miss, that would be about as complicated as a wild card clinch; most likely 9 losses is going to keep you out of the playoffs most years (but not 8), but theoretically a 11-5 team could miss the playoffs, so it's very hard to say what is "possible".