This has literally never ever happened or been advocated by anyone other than Islamist extremists.
Eh, what? At least in Germany there have been debates about that. There was that case where a domestic violence case was rejected because the judge thought beating your wife is a cultural thing for muslims. And there are so-called sharia mediators, who privately settle disputes between families, even about violent crimes.
* kaijyuu sees no reason to force anyone to learn the most common language of a place.
It only really harms them. And maybe annoys racists who don't want to hear people talk in other languages or *gasp* see signs/menus/etc written in other languages.
Also I'm not too well versed in Sweden's immigration policy, but I really doubt anyone can live off welfare without being a citizen, and I don't think they allow people to become citizens if they can't support themselves.
I don't know about Sweden either, but yes you can be eligible to welfare in other EU countries as a EU citizen and at least in Germany there are circumstances where you can get welfare even as a non-EU citizen. (That is kind of the difference between immigration debates in the EU and the US, here it's "immigrants live off our welfare state" instead of "immigrants take our jobs".)
The language thing is mostly about this: Many (muslim) immigrants marry someone from back home who does not speak our language. Thus their children don't learn it at home and now a 3rd generation immigrant speaks German worse than his grandfather. Also the spouse is unable to work, because you're not gonna find a lot of work here if you don't speak the language, thus the family gets welfare benefits.
You don't have to be able to support yourself to become a citizen, if you are born here. (Then there's also the law of return, but I don't think it applies to many people anymore). Though as I said, many residents are also eligible for certain benefits.