Should a generator transfer switch have continuity between all contacts in the "Gen" position?
If you are testing continuity at the generator inlet plug with the transfer switch in the GEN. position, you will be measuring continuity throughout the entire circuit.  If you look at this image, you should see why you're getting the readings you are.



The light blue box on the left, represents the main service panel. The light blue box on the right, represents the transfer switch.

Have the installation inspected by a certified licensed electrician before using the generator, to make sure the transfer switch is installed properly.  Remember, peoples lives could be on the line here (pun intended).  Back feeding caused by improper transfer installation, can kill linesmen trying to restore electrical service to your home during an outage.