How to request source code?
Whether common or not, it is acceptable to ask. Ask the main professor, who may pass on the request to a co-author. 

I'd suggest that you only give a small amount of detail in the initial request but offer to say more if you like. But don't flood the professor in the initial request. Mention that you have a draft, but don't send it. And you can certainly mention that you are interested in applying to the university. 

What happens next depends on the nature of the reply. Perhaps they will want to help you, but other outcomes are possible.