Change ammo as you should have 70% bare minimum in a 30" circle at 40 yards. I see 28 out of 41 pellets on the paper which is horrible for THAT full choke. You would have still put the coyote down on that shot for at least a follow up shot as you would have hit the coyote with probably 7 pellets, and 4 is enough to kill one with #4 buck.
40 yards would be your longest shot, right?
Here is the answer to the shooting low:
http://www.ebay.com/itm/Jack-west-adjustable-stock-Remington-1100-1187-trico-west-/161950612659?hash=item25b5023cb3:g:OYQAAOSwpRRWnFgO
Shotguns are a lot more complicated than you would have thought, eh? Even if you lay down $1400 for a shogun and start analyzing patterns the way you are, you sill will have to find the ammo the gun likes and deal with shotgun fit. An education costs money, time, and trouble.
Even if you sell the barrel and buy a rem choke barrel, you will have to go through the same issues with chokes. I have three extra full chokes for my Beretta 391 in extra full and they all pattern differently with the same ammo... Turkeys.
Your fit on a shotgun will probably mean that you shoot low with a lot of shotguns, so be prepared.
You paid $325 for the 1100 3", vent rib full choke with a good Neopreme sling and sling swivels, the action alone was worth that. So, if you are not happy, sell the barrel on ebay, as it has had the forcing cone lengthened cone lengthened, and an extra gas port added, drilled and tapped with a set screw so you can shoot light shells also. The barrel should sell for $150 on ebay, put the money toward a Rem choke barrel, and there is a Rem 1100 3" Hastings Wad lock rem choke type barrel on there right now.
Before you do anything else, you should try some Federal Premium and some Winchester Supreme 3" #4 buck, along with some 3" 1 7/8 oz BB.
It is too bad that things in life are not as simple as we need them to be, as job and our budgets seem to get in the way.
In an answer to your question in your first post as to whether you have a viable shotgun or not, consider this, in the picture below you see where the individual name tags that was taken off of many single trophies and were consolidated onto this one plaque. The Predator Hunting club asked the older members to donate their trophies to save money and give the new guys more of a chance to get nice trophies. I gave another box of trophies back to the club to recycle to new guys when I left Ca. That 1100 of yours was on everyone of those hunts. We would have not tolerated a shot gun that was knocking down coyotes with them running off costing us to loose a club hunt or State hunt.
That shotgun has killed over 500 predators of different kinds, and I will say it again, a shotgun that gimped up coyotes, missed foxes, and bob cats would not have been tolerated for one minute.
You need to work on two things, gun fit, and finding the ammo that shoots well in that shotgun, the factory has obviously changed the load that they loaded years ago. Nothing has changed about the shotgun except that I took off the camo tape.
For the kind of terrain(thick woods) that you hunt in, having a turkey choke shotgun would not fit your needs well, as these kinds of chokes are better suited for 35-60+ yardages in open terrain.
Good luck
The same thing will apply to your crow shooting, shells from different makers will vary greatly in their ability to pattern.