I was in Leiden about 2 weeks before the classes started and 1 week before the introduction week started. The introduction week was very fun, helped in getting to know other exchange students and learn useful things about the student life in Leiden. It also was very helpful in informing what we should do to avoid extra paperwork with registration. During the introduction week I received a Dutch SIM card for my phone, it's much cheaper than using your Swedish one, the reception gets better and you can use internet on your phone without being financially ruined.
It was only me and a friend of mine who went on the exchange to Leiden and on arrival we simply wandered around the airport until we found a help desk that could direct us to the train towards Leiden. Once we arrived to Leiden my friend got picked up by a girl living in his dorm while I got on a bus. In retrospect I can recommend speaking with someone in your dorm (if you live in one) to pick you up at the train station and direct you. Since I lacked this guidance I got lost and spent 40 minutes traveling with 50 kilos of luggage in the wrong direction.
When I finally arrived to my dorm I was greeted by a common room filled with Dutch speaking strangers. I introduced myself and immediately got a bear passed to me and everything was good. I felt like I had been there for a month already and that I was a part of the group.
I was told to go to town hall to sign up for a temporary town residence thing, but they required a copy of my birth certificate. I still filled in the forms but didn't provide my certificate, after 2 weeks I got a temporary Dutch social security number and there were no complications.
My other friend who traveled down with me didn't sign at town hall but still got his grades, I therefore believe that the registration is only useful if you are arrive from outside the EU or if you want to open a bank account in Leiden.
It might be good to get a bank account since very few shops accept Visa or MasterCard, instead they have their own system with maestro cards. You can live there without problem without a card, I did. But then you need to constantly carry large amounts of cash on your person.