Hei Mari,
Tervetuola!

I'm also working for a telecom company that has a presence in Finland. My company also made us sign a paper that sounds similar to yours (altho mine was only 5-6 pages, I think. So maybe not the same company

). Anyway, regarding internet at work, it's pretty relaxed (I'm typing this at my work

). I also see others using internet for private reasons. The phone bill is given to us at the end of the month, and we should seperate private calls from business ones, and pay for the private ourselves. But this is also relaxed, unless you have some really expensive private calls.
Any other questions, feel free to PM me.