Thanks for sharing your profile. Will try and address the questions I think I can help out with..
My Questions:
Is my profile too bizarre to live?
-Very bizarre profile and you will have question marks on career focus and obviously your GMAT, admission is not impossible to overcome but clarity with your future goals, fit with your target program, and explanation of past decisions will be critical
Will my behavior from age 17-21 forever damn me or will my GMAT keep me out of the discard pile?
-You are in rough shape with your GPA, a 2.7 is pretty low. No such thing as an automatic ding especially with a GMAT score as strong as yours this should signal you have the aptitude to compete in an MBA program
Should I load up on more extension courses these next few months to bolster my academic credentials? Will I get diminishing returns after one or two?
-extension courses are not the same thing as being officially enrolled in a sit down program so they are good and can showcase your ability to handle specific coursework but will not be seen as the same thing. So prioritize a few classes like acct, finance, stats that are key for MBA programs and then move on. They help make your case but your undergraduate record and to a lesser extend your masters will be the primary area of focus
Are MOOCs a good supplement to treat a bad GPA?
-they help but will not erase poor performance (See above)
I was thinking of applying R1 to 4 or 5 schools, and maybe submitting R2 to a Top 5 after I build up my alternative transcript. Is that advisable?
-Sounds like a reasonable approach, 4 or 5 is a lot for one round though, may want to balance things out a bit
How reasonable are Tepper & Ross?
-Tepper more reasonable than Ross but Ross tends to be a bit more understanding than many other top programs
Is Tuck a fair reach?
-Tuck will be tough but nothing wrong with a few stretch apps
H/S/W: Nah, right?
-Probably not, stanford is an interesting program though if you want to take your shot at one, this may be the program
Where else might I consider applying?
What about those recommendations? Is a good colleague better than a good professor?
-Depends on your distance from academia but i would say a colleague would be better