Question: This has puzzled me for a few years now. When preceded by *'a'*, *shop* becomes a noun. Does "do a shop" even make sense then? The correct phrase for me was always "go shopping", or similar. 

Can anyone tell me where "do a shop" came from and whether it's grammatical?

Take the article in the following link, from an Irish broadsheet newspaper:
<http://www.independent.ie/lifestyle/parenting/i-have-seven-children-and-my-weekly-shop-includes-200-nappies-and-20-litres-of-milk-3217903.html>  

The headline, which appears to be a quote(but is not contained in the actual article body) shows an example.  

Take the following line from the article *"it takes two trolleys to carry the weekly shopping"*. If we were to replace *'shopping'* with *'shop'*, would it still make sense?

Answer: 
I think it has a different and specific meaning.

"Do a shop" or "Do the shop" is a specific task, eg. the weekly trip to the supermarket for the big grocery shop.

"Go shopping" is a more general, recreational, go out and look at the shops but with no specific aim.

It's the same way that "load" is a verb or a noun. Shop is the place you buy stuff, the process of buying it or now the stuff you buy.
"Do a shop" is no more or less "correct" than other slang, e.g., "Let's do lunch", "Let's take a meeting", and "**Enhance** your manhood by 3 inches instantly!" It's what people say. If they say it, and if other people understand what they mean, what on Earth does it matter?