“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.
Focusing on the "bad" ...
In AmE:
A working class neighborhood is typically not considered a "bad part of town". These neighborhoods usually have relatively lower rates of violent crime and drug abuse, and the housing stock is usually well-maintained, with a relatively higher number of homeowners; trash removal by the city is regular, and there are usually nearby conveniences such as food markets.
A "bad neighborhood" typically has a high rate of violent crime and drug abuse, and the housing stock is poorly maintained, often by "absentee landlords" who allow their rental properties to become derelict; trash removal and other municipal services are less reliable and refuse can build up on the streets; there are few conveniences. These neighborhoods are often located in what have been called "food deserts", where access to fresh foods is limited.
"The boondocks" and colloquially "the boonies" are rural. Population density is lower. Violent crime rates are usually much lower than they are in urban "bad neighborhoods", but a good many rural communities have been hard hit by recreational drug abuse, methamphetamine and heroine. Sometimes the housing stock is ramshackle, but not necessarily so. Food markets are usually quite a distance away. These areas are rarely given over to subsistence farming. Trash removal is not handled by the municipality but by private contractors, and so in rural communities one often sees discarded items on the properties which would have been removed on "large trash pickup day" in a suburb.
"The projects" are government-run rent-subsidized dwellings, sometimes low-rise, sometimes high-rise, usually found in urban centers. Many of them have high rates of violent crime and and drug abuse and they are often found in or near "bad neighborhoods".
Best Answer
1) To me as a native BrE speaker all of those sentences are OK (apart from correcting "wonder" to "wondering"). Although "check out" originated as AmE it is now heard in BrE too in this context and others. I don't see much difference in the degree of formality. In my experience tradesmen and women are not much interested in formality as distinct from politeness. All the sentences are quite polite with the third being very much so.
If you were going to take a vote, I personally would not use check out, but I would use the third sentence.
2) Other terms: "house call" is associated, but not exclusively, with visits at your home by your doctor. A quick duckduckgo search reveals that there are lots of companies offering PC repairs "at home". Another term often used is "call out": sometimes you are offered service at home with no call out charge, other times you have to pay such a charge. In my experience different traders use their own different terms.