Will transformer operate a door bell/chime 200ft away?
Not a standard door bell transformer no.  #1 A standard door bell transformer is sized to drive one chime only.  If you want a second chime you need to design a system.  I recommend you call a manufacturer of chime kits and ask for a tech rep. Broan, Nutone.  Tell him your situation. He will tell you what you need.  A standard chime transformer are 16-volts.  I have used 24-volt transformers on longer runs with success.