Yes, it means romantic partner, wouldn't do for a business partner etc.. I'm guessing boyfriend is not accepted because בן זוג implies a stronger commitment than boyfriend, for that we typically use the word חבר which is ambiguous and causes confusion and embarrassment no end. Fun! :-)
Yes, it's wrong in this sentence, although, as MaryJaneKe4 said in a previous comment, some people might say it. It's wrong because the sentence tells whether each man works; so each man is a subject of the verb "to work" and the sentence means:
"His partner works, but he does not work."
The final verb "work" is implied, and "he" is the correct subject pronoun. We could even say:
"His partner works, but not he."
However, that has a formal sound that many people won't like.
Since "him" is an object pronoun, we could use the phrase "but not him" in a sentence such as:
"I will work with almost anyone, but not him."
(In that sentence, the ending "...but not with him" is implied.)
I disagree. As I stated in my comment elsewhere, this is a sentence I would use without even thinking about it. Since I am a native speaker, it is not wrong to say it. Unless you wish to maintain that something which millions of native speakers would say without blinking an eye is "wrong" because it does not correspond to a rule in a grammar book somewhere. Languages evolve. Grammarians often describe the state of a language from 50 or a hundred years ago; it takes that long for them to catch on, or accept, how people naturally speak in their own time.
"Ungrammatical" is a technical term that refers to whether something is allowed by the rules of a particular language/dialect, whether or not that dialect is standard. If something is heard, then it's grammatical, at least in some dialect. I would note that for me personally, "his partner works but not him" is grammatical, while the other examples you give are not. I don't think any of them (including the one that's grammatical for me) are standard.
It won't let me reply to your question (too many levels of nesting?), but a "standard" is a prestige dialect of a language that has been codified, i.e. has rules that are written down. Often these rules don't align with the spoken language, whether because they're completely artificial (English grammarians of the 18th century loved to "import" Latin grammar into English where it didn't belong), or just because they are rules that were true for an earlier stage of the language but no longer are. "Grammatical", in the sense of referring to a particular sentence or phrase as following the rules of a language, was specifically coined by linguists to differentiate descriptive observations about language as it is spoken from prescriptive rules about what should be used in the standard dialect. That's why I objected to using "ungrammatical" to refer to something that violates the rules of a standard even though, as you said "it could be heard", meaning that for some dialect at least (including mine), it's grammatical.
@brjaga - I agree with your basic idea, but the word 'dialect' may not be the most accurate to express your idea. The sentence "His partner works but not him" is spoken from coast to coast, and so unless you call 'American English' a dialect, the word dialect does not describe actual reality. -- The rules of a language, at least for American English, are not dictated down to the people, but rise from the people. The French have a body which can dictate grammar rules (l'Academie francaise), but even in this case the Academie has on occasion had to cede to popular usage. Usage is the final arbiter.
"His partner works but not him" -- this is correct in American spoken English. In talking to my wife about another couple, I could easily say this sentence without any hesitation. OTOH, "His partner works but he does not" would sound very stiff in this context, although I could imagine it in a textbook on English for foreigners. It might be used in literary English, but not in (my) spoken English (I am a native speaker of American English).
Gabriel Cavalcante (Malo665m3ntalbr): Yes, by itself, "בן" means "son", but it can also mean "boy"; similarly "בת" means daughter, but it can also mean "girl". For example, one may call the boys in a group "בנים" and all the girls "בנות".
Be careful, though, about how you use the literal meaning of individual words in a phrase. By itself, "זוג" means pair or couple, but we translate "בן זוג" as a "partner" (who is male), not as "son of a couple"; similarly "בת זוג" is a "partner" (who is female).
It's helpful to read the other comments on interpretations of the "partner" phrase.
In the sentence you typed, Yuval introduces himself, saying his name and that he is 7½ (years old). In both of those phrases, a literal word-by-word translation would not produce a good English sentence.
To ask him his age, you may say "בן כמה אתה, יובל".
To ask Yonit her age, you may say "בת כמה את, יונית".
And if she's 7, she'd answer "אני בת שבע".
If you were taking a grammar test and wrote “his partner works, but not him”, you might be marked wrong because you used “him” instead of “he” such as in the phrase “...but he does not”.
If you were talking to a friend and said “his partner works, but not him”, it would be an unusual friend who would correct you for being wrong even if you are “wrong” in a textbook sense.
“Wrong” can mean “does not adhere to traditional grammar” or wrong can mean out and out incorrect such as the phrase “I have two cat.”
Saying “his partner works, but not him” is wrong in the first sense but not in the second sense.
@ruckman - who are you to say a homosexual couple is not natural? Zoologists have observed same-sex couples in nature in a number of species. Is nature not "natural"??? It may not be the majority, but it exists. -- "There are more things in heaven and earth, my dear Horatio, than are dreamt of in your Philosophy." (Shkspr)
man, why are you saying that, gays are people like me and you, just because they have sex in a different way, it's not like they are not humans. There are a lot of gays in Israel, I don't live there, but I've known that Tel Aviv is one of the most popular cities of the proud gay. So you already should learn how to live with differences.