No, you can have the same subject for both main clause and subclause as long as you have a conjunction such as men, which is a coordinating conjunction. There are also subordinating conjunctions, like medan ('while'), and with those you can't: Jag sjunger medan jag äter 'I sing while I eat'. This works pretty much the same way in Swedish and English.
Actually in English it would be perfectly acceptable to state the subject only once while using while. "I sing while dancing" is just as acceptable to say "I sing while I dance".
Now that I've said it, there's a slight difference in meaning -- "I sing while I dance" implies a single event, whereas "I sing while dancing" to me says that singing is something I always do whenever I dance. Now I wonder if there's a specific grammatical rule I can point to, or if that's more colloquial usage?
Yes, but it only works with the gerund in English, and we don't have that. You can't do it with the present any more than we can: I sing while dance. The reason you can do it with the gerund is that in that case, the whole phrase functions like a noun. (I think: the English gerund always makes me a bit queasy. I've mainly studied Russian grammar, not English).
Native English speaker here. It doesn't seem to me that it is a gerund here, instead, a present-continuous verb with some implied words left out. For example, "I sing while (I am) dancing." 'I am' isn't spoken but is implied. Gerunds are nouns and don't really seem to fit here.
A small note here too. It seems to me that "when" is the more common conjunction, instead of "while", though both seem acceptable.
You're probably right about dancing. I said the whole clause functions as a noun, but come to think of it, it actually functions as an adverbial.
About when/while, it doesn't really matter which one is the most common since there's a difference in meaning. 'when' would be när in Swedish and 'while' is medan.