Hi Mikko,
Sorry to hear about the problems with your computer.
As there is no anti virus product installed on your computer at present, and you are unaware of how it may have been used before it came into your ownership, its well worth you considering restoring the machine to its factory condition.
This will have the effect of clearing out the operating system and software but also clearing any documents or photos or other files and applications, so do make sure your back ups are good, comprehensive and includes everything that's important to you on your computer (stuff you don't want to lose permanenly)
Once you have restored your computer give it a good test before you load applications on it... has it improved the situation?
Don't copy from your backups yet, as they could have been compromised by any malicious code that may have been present on your computer, only copy back stuff when its absolutely necesaary and then only after checking it with your updated resident anti virus product and at least 2 online scanners of good repute.
Post back and let us know how you get on.
Regards
paws