Is there any granular way to change decisions made in ME1/ME2 to see different outcomes in ME3?

In ME1 and ME2, I had to make decisions not really understanding their impact on the series as a whole. Now that I'm partway through ME3, I find myself curious as to what would have happened had I made a different choice previously. What if I'd done X? What if I'd let Y live/die?

I can't just replay ME3, import my ME2 save, and make different choices to see these outcomes - I'd have to make changes to the way I played ME/ME2. Likewise, importing no save would make wholesale changes to my experience that I'm not sure I'd appreciate.

Besides starting the entire series over again, and running through each game dozens of times to explore the various permutations, what can I do to change the way things play out in ME3? Do I get the option to edit the imported starting state on New Game+?

I'm on the Xbox, so save file editing or importing from the internet is likely to be at least challenging, if not impossible.

Theoretically your best bet would be using a save game editor to change the flags regarding those pre-ME3 decisions and outcomes. There was one for ME2, but I admit I haven't checked if ME3 has one.

Alternately, I recall an archive of all possible ME1 variant ending saves that people could import into ME2 to get the start they wanted, hang on..

Bingo.

Your second best bet would probably be to use that site, download the save you want to import, play through, download next save, etc.

Do I get the option to edit the imported starting state on New Game+?

Unfortunately the answer is no, you can't change any of the previous outcomes when doing new game+.

Since you are on the xbox, a saved game editor is probably out of the question.

There are a few ways to transfer files from other xbox owners though for ME1->ME2:

To import save games on the Xbox 360, the game must be played on the hard drive the saved game was completed on, since it is not possible to transfer the file that Mass Effect 2 imports using the Xbox 360's file management tools (although it is possible when transferring the entire contents of the drive using a transfer cable). As it is possible to transfer regular save files using the Xbox 360's file management tools, the player may choose to transfer a save file right before the final battle and complete it again in order to transfer the completed game file. This will need to be repeated for every playthrough the player wishes to import. Alternatively, run Mass Effect 2 on the old Xbox to import the character, then save the game and move the resulting save file to the new Xbox.

I'd imagine the process is the same for ME3.