35v reading on 24v transformer
Doorbell transformers have poor voltage regulation (they're often Class 2 Impedance Protected), so an open-circuit voltage of 35VAC RMS is not especially alarming. Check it under load and it will probably be more like 24V. 

Don't short the doorbell button out to try it though if it's an electromechanical chime, they're not designed to handle being powered continuously and you can burn the coils out. A few seconds is okay.