The broken piece off the stock of a Flues right behind the top edge of the receiver is very common. Just a bad design. Nothing in there to really draw the action back into the wood when the screws are tightened up.
..and the top tang screw is a simple wood screw as already pointed out.
As it isn't the best of securing methods for the top tang, many have failed over the years and have been 'repaired' by placing a machine screw in it's place with a nut in the wood to thread in to.
If fixed in that way, the nut is inletted or otherwise set into the wood on the underside of the area opposite the tang.
If it's not inletted tightly or glued into place, it can turn with the screw as you try to remove it.
Luckily, you don't need to remove that screw to take the trigger plate off.
Take the rear tang screw out after removing the trigger guard,,and the trigger plate screw on the bottom out,,,and the trigger plate comes off.
Sometimes a gentle tap or two with a dowel from above through the opening in the table helps separate them.
Then you can see if there is in fact a repair to the top tang screw by looking up from the bottom at the area of wood opposite the top tang screw inside the stock.
Or in the very least you can push upwards on the point of it if it's broken through the wood to push it up and out.
The frame/receiver comes off the stock after that.
That area of wood is about 5/8" thick at most and has a cut out in the front area of it for the top lever spring and screw and at the rear for the safety spring. A prime place for a large wood screw to crack and split the stock.
The sear springs are attached to the trigger plate, so they won't be taking a ride on you when the trigger plate is taken off, and the sears can usually be left in place on the frame when removing the action from the wood. If they do interfere, drive the sear pin out and the 2 sears will drop free.
The sears go back in with the sear arms toward the center of the frame.
