Ryan: You can look here on the PGCA Homepage for a lot of wise information that touches on your questions.
https://parkerguns.org/ Look for “Parker FAQs”
Congratulations on receiving a quality firearm.
Any Parker Brothers shotgun (in safe shooting condition) is going to be a fine game getter.
The pad may not be cheap. But unless a very special order, it is likely a replacement and a replacement that has some age and antiquity of its own.
Assuming the chokes are not altered, you can expect them to be modified and full. The standard configuration for the Trojan grade. There are drop in gauges that can give you an estimate. Gunsmiths invest in the tools that can give you a more accurate reading on the Parker Gun chokes. The only way to really know what your chokes are doing for any given gun and load is to actually shoot at paper and then count the pellets.
If this were mine, I would get this to the best quality gunsmith I could who has experience working on the Parker gun. The alignment of the screw head slots is solid evidence that someone else has already been in there and didn’t quite know how to put it back together to the same way they found it. The protruding screw you refer to was probably misfit at the same time. Not fatal mistakes but cosmetics well worth correcting.
You don’t mention the gauge, I’ll assume it’s a 12 gauge.
Can you tell us the serial number, or post a better picture so we can make out the first three digits?
And the barrel length is …? How about top view pictures of the end of the barrels?