re: the stock,,I'd clean up the head of the stock to remove any oil. Avoiding removal of any of the orig finish from the checkering on back,,that's looks like it's in nice shape.

Then splice in a piece of wood for the break. Make the break line a nice straight line to make matching up a piece of repair wood easier.
Makes the glue line disappear easier and the joint stronger.

Match the grain as best you can and with some artistic help from the finishing processes you can have a repair that is barely noticable if at all,,especially when you add a bit of wear and 'patina' to the area to match the rest of stock.
No need to re-stock the gun at all. Saves an awful lot of work and you learn a lot too.

I would re-bed the action while at it w/ glas compound to make sure it's secure and further cracking doesn't occur. No need for any of that to show from the outside.

As to the bbl fit,, if the bbl breech face is flat against the action face,,that's where you want it. It looks good in the pic, but that's all I have to go on.

Unless the bbls themselves slide or 'shuck' from side to side that small amt of space indicated by the rib extension, I'd just assume the bbls were probably that way from the factory on the low budget shotgun that they were. Don't take that last statement as a knock against the gun. I've had and have plenty of these!

Moving the bbls and or frame around at this point if everything is on face now will probably result in being off face when you're all done.

If the small sliver of space on the right side of the rib extension bothers you, you can soft solder a shim to the side of the rib ext and fit it to the frame for a close(r) fit. It will only be cosmetic anyway.

Mild steel shim stock mat'l is avail in packets containing various sheets in thickness from .001 right up to .035 or more in the same pack.
Use a feeler gauge to see what you need in there, back off a couple .000 for solder and clearance and clamp and sweatsolder a piece tightly to the side of the rib extension. Trim and blend it in and the small void dissappears. No stress on the shim so the soft solder is fine for application.

The over sized frame breech on the left side could then be filed down to better match the bbl contour also.
These would satisfy the cosmetic problems you're seeing without doing major work.
Putting a faux patina finish back onto the filed and polished recontoured frame breech can be done w/o any problem.

Just some thoughts for fixing up project guns like this,,