As for the first part of the question, I do not believe Allen would have been there at 12 if we just stayed put. It's possible Rosen might have been, since Arizona might have traded up and taken Allen leaving Rosen. It's also possible that teams like New England or New Orleans might have traded up to 10 or 11, but couldn't put together enough to get to 7, so as the guy keeps falling, more teams are now willing to pay the smaller price it takes to move up and get him. Or possibly Miami liked Allen but didn't like Rosen - if Miami offered their 1 and 2 to TB, both picks being better than our 1 and 2, we would obviously have to beat that price to make the deal. Maybe Miami didn't make that offer but TB told us they did - then Beane got snookered. Or maybe TB heard that this was the offer we gave to Denver, so if we were willing to give up that much for Allen to Denver, why not give the same thing to TB for Allen?
Did we overpay by giving up 2 second rounders? I think that is a yes, but some times you have to do that. A lot depends on how enthusiastic they were for Allen.
These were the possible options the Bills could have offered: (point values come from a commonly referenced trade chart)
Both firsts - Pick 12 for 1200 points, pick 22 for 780 points, total of 1980
1 first plus 2 seconds - Pick 12 for 1200, pick 53 for 370, pick 56 for 340 - total of 1910
1 first plus 1 second plus 1 third - Pick 12 for 1200, pick 53 for 370, pick 65 for 265 - total of 1835
1 first plus 1 second - Pick 12 for 1200, pick 53 for 270 - total of 1570 - this would have been fair value at this point.
Here are the points needed for various picks
4 - 1800 points
5 - 1700 points
6 - 1600 points
7 - 1500 points
It has been reported that the Bills had an agreement with both the Browns and the Broncos for a deal - at a minimum, that would have required at least our first pick in each of the first, second and third rounds to get Cleveland to say yes. I doubt Cleveland would have considered any deal that did not include both firsts - they already have a huge pile of picks, so no need to stockpile more in the later rounds.
Basically, I think Beane had arrived at the conclusion that he would be willing to give up 1 first and both seconds to Denver in order to get Allen. Once he decided he would give up that much for Allen, don't think of it as trading for a "pick" but trading for a "player". Presumably, we would not have given up that much for any other elite player on the board like Chubb or Barkley, even though the point value would have been the same. The Bills need for a QB, coupled with the Bills determination that they liked Allen a lot more than Rosen, forced them to pay up. And it's possible they liked Allen better than any QB, including Mayfield and Darnold - neither of whom are sure-fire can't-miss prospects either. So if they were really getting the best QB in a QB rich draft, then that would be a small price to pay.