Learn English – How to answer “What would you like to drink?” in this case

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.

In OP's exact context, a better phrasing is...

"The drink bought by my friend was an unpleasant surprise"

That's if we assume OP (or his friend) experienced the "unpleasant surprise" immediately upon seeing or tasting the drink. If it was, say, some kind of powerful delayed-action emetic that had no apparent effect until several hours later (when it caused sudden unexpected projectile vomiting or worse), you might perhaps say the drink "led to/caused an unpleasant surprise". But that's a highly contrived context.

Note that bought is past tense buy = purchase, so OP's phrasing is ambiguous in that we don't know who the drink was brought for (or by implication, who got the unpleasant surprise). More natural phrasing (optionally using the past tense of bring = carry/give [to someone]) would be...

"The drink my friend brought/bought me was an unpleasant surprise"