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: 
The OED’s definition 2e for *shop* is ‘An act of shopping for purchases’ and it is described as colloquial. The earliest citation in this sense is from 1960: *You should find it possible to have one big ‘shop’ a week with a small mid-week ‘shop’ for perishables.* Once we see *shop* used as a noun in this way, any grammatical difficulty over it following *do* disappears. It's fairly common in the UK, but *shop* is more likely to be preceded by an adjective such as *big* or *weekly* than not.
"Do the shop."

Usually means, "Do the weekly shop".

Grammatically, going for a shop, is like going for a walk. "Walk" can be either a noun or a verb - "to walk", "the walk".

When people say, "to go for a walk", what they mean is "to go walking" or to go on "the walk", or "the usual walk".

Do people stop to consider whether it is a noun or a verb before they say it? I don't think so. Is it understandable? Yes

Possibly it is an example of an ellipsis: 

> 
> (grammar, rhetoric) 2. The omission of a grammatically required word or phrase that can be inferred.
> 
> 
>