Hi,
Just to confirm what Tuna said - the shutter will be 1/500 without a battery - you can manually adjust the shutter, so some results are possible if the light is right.
Without a battery, the metering just does not work and this symptom can often be seen if the lead at the top of the battery container has rotted off its terminal (happens quite often), but you should see that the battery test light is intermittent, or non-functional.
The Pad of Death is a small rectangle of rubber between the primary and secondary shutter release shafts, inside the body. If it has rotted away, or gone sticky like toffee, this will throw the metering off and you can get a variety of symptoms. These range from the shutter never closing, the under expose light staying on during exposure, and only getting 1/30 shutter speed (plus others). It is fixable, some have done it by only removing the top plate, but I prefer to take the front plate off - this way you can make sure that you get all the bits of a crumbled pad out, ensure the sliding contacts that operate the metering are clean (I have had one where one of the fingers was bent) and replace the light seal around the outside of the rear lens element.
There are lots of sites around that cover this, and it is not as hard as it looks, providing you are careful and methodical.
Hope that helps