Learn English – Going to vs to go to

The quick answer: In most contexts, these two words are effectively synonyms, especially when discussing possibility, e.g. "(Can/Could) you [perform some action]?".

Longer answer:

There is a somewhat subtle difference in these two phrases that becomes a lot less subtle in different contexts. Let's take the words one at a time:

• Can deals with actual, literal ability to perform an action. When you ask if someone can explain something, you are literally asking whether they are capable of explaining it. Edit: It is important to note that, as @J.R. has pointed out, this is by no means the only use (or even the most common use) of the word can or the phrase can you .... This construction is often abused to mean effectively the same thing as could you..., i.e. would you be willing to..., etc.

• Could deals with the possibility of performing an action. It is definitely a subtle difference, but basically, when you discuss possibility, you generally are aware of their ability to do it but are tentative about their willingness or availability to perform the action. This is by no means an exact definition.

It is also worth noting that the word could is actually the past tense of can. In English, using the past tense form of a word in this way is often meant to indicate tentativeness.

I wanted to talk with you.
I wondered if you would be available this afternoon.
I thought it might a good idea.

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.