I heard it comes from the Egalia kindergartens. Is it used colloquial or only in texts/officially?
It does not come from those kindergartens, but they may have played an important part in making it popular. The word was first proposed by a linguist in 1966. In the 90s and 00s it started to gain ground in queer circles and from there it spread further. Egalia didn't open until 2010.
Today the word can be found even in official texts like legal proceedings, various texts from municipalities and companies etc. It remains controversial however. See some native speakers disagree over it here: https://www.duolingo.com/comment/5531330
Does this mean "He will only sleep in a bed with sheets that are clean" or does it mean "He will only sleep on top of clean sheets" (or would the latter be "på"?)
Sleeping ON top of clean sheets, would always be PÅ. I rena lakan is always between the sheets, 'inside' the bed.
Since rena has the -a ending and lakan isn't in the definite, you can tell it's plural: the sheets.
I wish Duo would explain things like this without having to read many comments to find the awesome ones!
At least, on mobile..
We have lesson notes that explain these things, but for whatever reason you're not allowed to read them on mobile. :( Rest assured we're just as annoyed by this as you are.