Learn English – “available in the store” Or “available in-store”

Client and customer are no longer distinguished in the way they once were.

As recently as fifty years ago, client was still by and large reserved for people who sought advice from ‘professionals’ such as attorneys and architects. At that time client still reflected, even if only very dimly, the original Latin sense of the word as “one who is under the protection of” a patron. A client was someone who benefited from the professional’s learning and knowledge, while a customer was someone who had a purely commercial relationship with a tradesperson.

In this context, the term client acquired an honorific character. It flattered the person rendering services because it imputed a ‘professional’ degree of expertise; it also flattered the person who purchased the services because it imputed a degree of wealth and status—a person who routinely consults professionals is clearly someone above the common run of customers. And it happily set both at some distance from the vulgarity of commerce.

In consequence, from the last half of the 19th century, businesses of all sorts started referring to their customers as clients—‘exclusive’ tailors and hairdressers, grocers, insurance agents, even bookmakers (now calling themselves ‘turf accountants’). And since it is mostly businesspeople who talk about the relationship between buyer and seller, that use has gradually extended into ordinary speech.

The upshot is that today there is no difference whatever between customers and clients—they are the same people. Instead, customer is used when you wish to focus on the financial aspect of the relationship, client when you wish to focus on the personal service rendered.

I think in some ways you have answered your own question(s).

1. Yes, "go shopping". One would not say "go storing". Shopping is, as far as I know, used the same way in UK and US English, and has nothing to do with "shop" vs "store".

2. and 3. You would either name the store or type of store ("I'm going to the hardware store") or, as you say, ""I'm going shopping" or "I'm going downtown", but not "I'm going to the stores". You might be inclined to add specifics: "I'm going shopping for a hammer", for example.

In certain places, you may find "shop" and "store" are interchanged (I certainly do this, having lived in both US-English and UK-English countries). If you want to sound natural, though, stick with what is usual for your area.

Of interest:

eytmonline.com says that "shop" meaning "booth or shed for trade or work" is c. 1300, while "to visit shops for the purpose of examining or purchasing goods" is from 1764 (no idea what they would have said before that!).

Meaning "building or room set aside for sale of merchandise" is from mid-14th century, while "store" for "place where goods are kept for sale" is first recorded 1721, in American English.

There was apparently a strong distinction between "shop" and "store" in US English, with "shop" having retained its original (1300) meaning:

The word store is of larger signification than the word shop. It not only comprehends all that is embraced in the word shop, when that word is used to designate a place in which goods or merchandise are sold, but more, a place of deposit, a store house. In common parlance the two words have a distinct meaning. We speak of shops as places in which mechanics pursue their trades, as a carpenter's shop a blacksmith's shop a shoemaker's shop. While, if we refer to a place where goods and merchandise are bought and sold, whether by wholesale or retail, we speak of it as a store. [C.J. Brickell, opinion in Sparrenberger v. The State of Alabama, December term, 1875]

Not sure whether that use of "shop" remains in US English.