What alternative designs competed for the C-130 in the 1950s?
The competing companies (ones that elected to compete anyway, more were invited) were Lockheed with two submitted designs, Boeing with one design, Chase Aircraft with three designs, Douglas with three designs and Airlifts Inc with one design. As far as I can tell, these were just designs and Lockheed won the contest based entirely on paper. The YC-130 was ordered and after performance was verified, it went into production in 1954. 

It doesn't look like there is any public information on the competing designs since they didn't leave the drawing board. It is unlikely that any of them are "clean sheet" designs. The Boeing submission was probably (and this is a personal guess) based on the C-97 Stratofreighter. The C-130 itself looks to be an up-scaled version of the Fairchild C-123 Provider with design elements from the Chase XCG-20 (which was developed into the C-123). 

I'm not sure what you mean by "justifications for the Lockheed design". Given that it is the United States military's most successful aircraft design ever, with 60 years of continuous service, I'm sure every design was driven by a contract or performance requirement. Here is a good article about the C-130 designer, Willis Hawkins that gives some insight into the project.

Most of this information was taken from the Wikipedia page on the Lockheed C-130 Hercules.