I think in some ways you have answered your own question(s).
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".
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.
I think they're all grammatically okay, but the second two constructions are very unusual. Such a phrase is normally constructed like this in colloquial conversation:
Who are you, really?
Who are you, actually?
These questions would be asked by someone who has suddenly become aware that someone else is not who they previously said they were.
Who exactly are you?
This construction is common involving the word "exactly" specifically. I've seen the other construction used with "exactly" as well. The difference in meaning is slight (and the three sentences could likely be used interchangeably without anyone noticing) but this phrasing suggests more that the speaker is merely confused, not deceived. For instance, a story about the American comic book hero Superman might show him in his civillian identity, lifting a car over his head, at which point an amazed bystander might ask, "Who exactly are you?" The word "exactly" suggests that the speaker had a vague idea of who the person was before, and that needs clarifying. "Really" and "actually" suggest that the person was falsely someone else, since we are now asking for the "real" or "actual" identity of the person.
Best Answer
Both solutions are possible in the first sentence. Both of them mean the same thing.
However, with the second sentence it is grammatically correct to add the preposition "of" after are:
It just happens that the word you picked for this example, "designs", has multiple meanings. "Design" can refer to either the scheme for how the components of the watch will work, or a decorative pattern applied to the watch. So if it is used with have, the meaning of the sentence would depend on the context. If are of is used, it could have only the first meaning.