The bit on your diagram that says relay should actually be the relay contacts. The relay coil would be connected to a transistor and then to your arduino. Like the first diagram here:-
http://www.thebox.myzen.co.uk/Workshop/Motors_1.html
I read that the small Christmas lights are 2.5V.
No it depends on what type of lights they are. I am assuming you mean filament lights not LEDs
However, if it were me I would go to a thrift shop and get a cheap string of LED lights that could be operated off 12V.