This has got me quite convinced that Fred and Fred is going to happen. They are probably connected magically, rather than acoustically, so they might be able to communicate across time. This setup might create the time beacon Harry was wanting.

Or, maybe their connection does not link through time. Send a pair of Weasleys back in time. You now have 4 Weasleys. Wait not-quite-an-hour, and then send 4 Weasleys back in time… 4 Weasleys is twice the number of Weasleys. Are N Weasleys N/2 times as smart as 2 Weasleys? No. It is much more interesting if it is the connections that matter. HE is the Weasley hivemind.

This chapter showed that, if it appears that a Time Turner wasn't used, they don't try to use it. Presumably, the reverse is also true. If it appears that it is used, they use it.

I've always figured that the rules deciding which stable time loop were something along the lines of the more likely it is for an event to cause itself, the more likely it is that one happens. If you want a specific time loop to happen, such as giving yourself a paper that factors a given semiprime, you'd make it so that happening causes itself, by copying down the factors if they are correct, and make it so it not happening causes a paradox, by writing something else down. This way, a high portion of time loops are the ones you like.

That can't happen here. They try to cause whatever happened. This means that any stable time loop that isn't too difficult to carry out is equally likely to work. It's implied that there's some sort of force at work here. While it's conceivable that most of the stable time loops with Harry factoring a semiprime were in the same reference class as "DON'T MESS WITH TIME TRAVEL", Dumbledore later managed to use the effect to gain actual information: the time travel he was about to attempt shouldn't be attempted. What's interesting here, though, is that it seems to imply that this force isn't just something that manifests when they do something wrong. It always chooses the time loop. Or more accurately, the rules that I had assumed worked whenever someone wasn't messing with time travel never work. The theory was completely wrong, instead of being something that breaks down under odd circumstances.

And I still wonder: why did the force let Harry do everything he ever wanted to do with time travel before, but then stop him now.

That force just made a very dangerous move. Perhaps it's not trying to do anything like paradox avoidance, as the "DON'T MESS WITH TIME TRAVEL" suggested. Perhaps it's trying to avoid all future paradoxes, by making Harry end the world. I've read about one story where attempting to abuse time travel resulted in the sun going nova. This is might be the same idea, but on a larger scale.