Trulia lists US properties. There are no international properties listed here. You might best be served searching the net again under Stockholm, Sweden, to find the right property websites for that country. The real estate agents there should be able to answer your question.
Perhaps a call the the appropriate embassy (nationality/passport of the person wishing to make this move) in Sweden can provide the answer to your question about emigration requirements, approved length of stay, etc. and whether proof of same is required to rent/buy properties there.
Not that you asked, but it might interest you to know that in the US you don't need to be a citizen or even a resident to purchase properties here. In almost all communities there is no check of passport, greencard or any other credentials to determine if you have entered the country legally. Therefore, if you are in the US, you can generally rent a property. However, real estate agents representing landlords and rental management companies will normally require a rental application listing employement, etc. which they will confirm, as well as a credit check.
Realty Executives Exceptional Realtors
54 Woodport Rd
Sparta, NJ 07871