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.
“You seem forced” can be used to indicate what you’ve expressed
Your meaning may have been gotten better than you think. “You seem forced” employs a fairly common usage of “forced”, indicating strain even though there is not a physical force. Note that “you look forced” in your question’s title is a little different, having to do with the way that “seem” refers more comprehensively to apparent behavior and “look” refers more to physical appearance.
Another way to say this that would be specific to tone of voice, diction, etc. would be “you sound forced” which could be hedged a little as “you sound a bit forced” and would be helpfully followed by a bit of explanation or a question, such as “is everything OK?”¹ or “do you feel you have to talk to me?”
Expanding on the “look”/“seem” difference in “you don’t look yourself”
With “you don’t look yourself”, it seems like you are saying that something about the other person’s physical appearance is unusual compared with historical data or some assumed expectation (e.g. someone’s face is green or streaked with tears and you assume they are not always that way, even if you don’t know them).
If you were to say “you don’t seem yourself”, your comments would be understood to refer to someone’s overall presentation. This could certainly include physical appearance, but is more often used to refer to someone’s speech, body language, and other aspects of behavior. I believe this phrase would fit the situation you described.
Note for “you don’t seem yourself”: Similar to “you don’t look yourself”, you would be understood to either be making a comparison to ways you’ve seen this person act before or ways you assume they normally act. For example, you think that the person doesn’t normally put so much emphasis on their words or deliver them in a staccato rhythm.
The note from the end of §1 of this answer applies here as well. Since you’re guessing at something about another person, it probably wouldn’t be a bad idea to describe a little more what you mean by what you’re saying. I personally think it would be appropriate to ask a question aimed at learning more about how this person is feeling and/or what they think about the situation.
Best Answer
I like them all, and they are basically interchangeable. Everything I say in the remainder of this is subtle. "Grab me a cup" is a bit informal. "Get" is somewhere between informal and neutral, and "bring" is somewhere between neutral and formal. To my ear, "bring" sounds a little more polite than the others as well; it's definitely the one I would use with my boss or with someone I didn't know too well.
If it were someone I knew better, I think I'd use bring if I were making an imposition on the person (asking them for a genuine favor) and grab if I were implying that I was not asking for much. ("Can you please bring me a cup of tea" vs. "While you're in the kitchen, grab me a cup.")