Is the landlord of my rental home responsible for mouse extermination?

You live in a colder climate. You can EXPECT mice to move in when it is cold outside. Accept his offer of traps and use them properly. If he hires an exterminator, that is what THEY will do. Why would you insist on an exterminator to show up and place traps around the premises, when you can do the same yourself ? If you are thinking that an extermination service can miraculously make all mice go away, think again.