This is my first time building an eLearning and I have a question regarding using a LMS. Do you really need to use a LMS for CourseLab? I will be building a website (using Dreamweaver) and using CourseLab for the eLearning which will include quizzes, games, exercises. So, is it necessary to use the LMS and which one would be the best?

You only really need to use an LMS if you want to store and track the users progress an scores.
You could build with a site with a database back end and track users by name. If you want or need detailed info and much more granular control of users activities then an LMS is the way to go.

This is a good place to look and try different platforms is http://www.opensourcecms.com/