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You can marry your fiancé in Sweden, but you need to submit a number of documents. The list includes: a valid passport, a certificate of birth, a certificate of no impediment and documents that prove that both of you are free to marry (a single status affidavit - apparently that's different from the certificate of no impediment). The source I found says you have to go to the local Tax Office (Lokala skattemyndigheten) and ask them about the documents you need. They are probably the best address to go to find out about the exact procedure.
If you get a definitive negative result on your asylum application, you will most likely have to leave the country or you might get deported. Going to Belgium results in an illegal stay in Belgium. So that doesn't seem to be a solution that works in the long-run.