There is a tradition of thoroughly investigating a colony petitioning for membership in Tau Kappa Epsilon, and having an installation banquet. These traditions have been followed without exception since. When did these traditons start and with what chapter?

I would say the first investigation would be on local at Millican wanting to become a Teke chapter. That the banquet began with their acceptance as the second chapter of TKE.

You are correct April 19, 1909 the beta chapter.

In 1921, a group of men from a local fraternity at Ohio State Universitywas headed toward a TKE Conclave to petition Tau Kappa Epsilon for a charter when they picked up a cast-off horseshoe as a good luck token. Where was that 1921 Conclave held?

a. Bloomington, IL
b. Milwaukee, WI
c. St Louis, MO
d. Madison, WI

The TKE horseshoe
 

I had the privilege of dining with frater George Kirk and Tex Flint
at the Beta Rho (Akron) chapter's 10th Anniversary fete. Tex had
not seen Kirk since, as an undergrad, he picked up an old horseshoe enroute to MADISON, WISCONSIN for the TKE Conclave in 1921. Kirk belonged to a local fraternity at Ohio
State and they were petitioning for a charter...which was granted.
Tex Flint was the Grand Prytanis then. Oh, what was the question?

Who were the expansion leaders?

I do believe the answer is:
Dr R.C. Williams
Leland F. Leland
James C. Logan
Frank B. Scott
Bruce B. Melchert

All of which were Past Grand Prytani

Correct!

In 1946, Tau Kappa Epsilon penetrated the South for the first time when , a small predominantly Southern national fraternity, announced its dissolution. Five of the then eight active chapters affiliated with TKE. What was the name of that fraternity?

Alpha Lambda Tau....that is how we got Doc Williams (RC)

That is correct!

I was wondering where the 5 chapters are and what happened to the other 3.

Alpha Lambda Tau
 

There were 21 chapters chartered from 1920 to 1935, and
Oglethorpe was Alpha. The ones which became chapters of
TKE were Beta, Auburn; Delta, Louisiana Tech; Zeta, North Carolina State, Tau, Maryland, and Phi, Missouri Mines.
Check the Baird's Manual for further info. I do not know what
happened the the other three.
A good rush would be a higher priority than finding out what became of the other three ALT chapter, really.
Erik P Conard, TKE trivioligist

Who actually wrote the speech "Oppurtunity out of Defeat", given by Wallace G. McCauley?

I think Frater McCauley wrote his own address. TKE.org doesn't say anything to the contrary and mentioned things that he "said". So i'll stick with that.