Is this a doorbell transformer? How to test if faulty?
Yes, your method is sound.  

Another method is to momentarily touch the line 3 and 4 together, and if good will have a tiny arc. This can be done easily at the door bell button. Once you remove the button off the wall undo one of the lines and momentarily touch it to the other line. 

Usually the button will go bad or the chime.  By bypassing the button as described above, you're one step closer.