Sure!
(1) I had a surgery during my high school and another one during the first year of my college. As a result my first year GPA was way below par and it had an impact on my overall GPA as well (its 7.4 on a scale of 10 from a low of 5.5 in the first semester). Should I try to justify/explain/use this instance anywhere in my application?
Maybe. It depends on what else you might write about in the optional essay. But it's certainly something you CAN write about.
(2) Columbia MBA has the option of early application. However, they take a written commitment that you will withdraw all your applications to other schools and join us. This is definitely a great program, but I am also banking on the MFin from Princeton. So essentially it comes down to MBA in Columbia vs M.Fin in Princeton. Given my goals would you recommend that I apply in early decision (which has its obvious advantages) or go by the regular route.
I think so. It's advantageous. If you get in, and you decide to take the Admit from Princeton, should you get one, then you lose the money.
(3) Also, some programs have a rolling admission. I have given a stock exchange certification test for which the result is expected in a month. Should I wait for the result or apply right now?
Do you mean to Columbia? There's no advantage to applying RIGHT NOW. Even with their rolling admissions. End of July is totally fine. But you CAN use the next month honing your essays into weapons of destruction.