After four years in Sweden you would be granted a permanent residence, and one year later you could apply for Swedish passport.
The requirement of 'degree in computer science' is just a part of standard job description, at least I was never asked about my (missing) degree.
One interesting option to consider are 'consultant agencies'. They hire you on a permanent basis (often offering relocation) and then offer you to companies. This way you may reach companies not offering relocation themselves (e.g. startups). A small consultancy I am working for now, Go See Talents [2], is offering relocation from outside EU (along with mentoring for less experienced developers, which is probably not the case for you)