Hello.
I'm afraid I cannot give you a very comprehensive reply, as I'm only a 5th year student myself and don't know that much about residencies. But I can offer you a couple of pointers.
If you are doing a residency, you need to arrange it with the hospital. Try to contact the secretary who who is in charge of these things.
I have seen quite a few foreign people doing residencies or parts of them in my university hospital, but they have almost all been in anesthesiology. The reason for this might be the fact that when dealing with patients you would either have to speak Finnish or have someone translate for you. Most of the younger people speak excellent English, but the older generations may not understand it at all. For this reason I would imagine getting a residency in GP would be more difficult (and not as rewarding to you). Also, in pediatrics you should find more possibilities in the research section, if you are into that.
As far as internships are concerned, I think the system here in Finland is a bit different and they don't work the same way as in most of Europe. Basically, here you work as a doctor during your studies, starting after the 4th year, and after graduating you immediately become a fully legalized doctor, skipping the internship period altogether (or you could say you do it during your studies). This has changed very recently. I believe in the EU you need to complete the practical part of your internship before you can apply for a specialist position in another country, even if you are a legalized doctor in your home country. In Finland this means working experience of at least 9 months in primary health care and at least 6 months in a hospital. I don't know if this applies to residencies though. But I don't know if doing an "internship" is even possible in Finland; if you work here you will have the responsibilities (and benefits) of a legalized physician.
I am actually abroad myself at the moment, which makes getting more information a bit difficult. Which city were you thinking of applying to? I don't study in the capital city so I don't really know about their systems (though I have friends there), but if you plan on heading north I can try and send some emails to people and find out more. You can also send me a PM if you don't want to post the details here.
Hope this helps a bit.
~Sam
PS: There's one more thing I forgot to mention. I think in many countries hospitals rank doctors applying for a residency position based on their grades during medical school. In Finland the residency program works on first-come-first-serve principle and no ranking of applicants is done. So you might also want to check what the situation for residency spots in different hospitals is. I have heard you may have to wait a couple of years for a residency in pediatrics in the most popular hospitals.
