screws
The situation
There is a screw that is binding two metals together. The screw were being fast drill in order to remove the binding. Due to the fast drill, the screw is now harden. So, how to satisfy the following requirements
Remove the harden screws or destroy it completely or destroy the binding.
Preserve the two metals as much as possible
Illustration
It is highly likely that your beams are made of pressure treated wood. If so, you should try to use the same type of wood in repairing them.
SHIM - When there is a slight gap between a beam and a joist, that can often be cured with shims, slim pieces of wood (preferably of the same type, or if not available, a rot resistant type such as cedar). Shims should be forced in (tapped in with a hammer) so they are tight. Often you can drive shims from opposite sides. Since they are tapered, opposing shims can make a tight level gap filler.
LEVEL - If you cannot shim them adequately and need to move the joists to meet the beams, you need to make sure they remain reasonably level. They may have a slight pitch down away from the house, and that would be ok. You do not want to cause them to shift so that they slope toward the house. In some circumstances, that could lead to a drainage problem.
FILL - The holes in the beams can be filled with slivers of pressure treated wood that are coated with exterior glue. Force several narrow wedges in to pack the holes. Let the glue dry and then saw or chisel the surface flat. You can then drill new holes even close to the old ones.
The birch posts in the original design were made a certain length to serve a particular function. I am sure they were selected on the basis of strength, stability, and durability. As a first order analysis the idea to cut them in half seems to defeat almost all the reasons that they were designed for in the first place.
It will not be possible to simply cut the posts in half and re-join them in a manner that will restore them 100% back to the original look and function. That said there may be some ways to come close to that as possible if one can look past the cut lines and some fasteners / bolts.
There is some missing information that will be needed to provide the overall solution and/or suggestions. I can provide one possible idea for the case where these posts are deployed in a vertical orientation and the girth of the post is substantial.
The idea is to go ahead and cut the post in half and then install a heavy piece of steel pipe into a hole drilled into each end of the post. The pipe would be glued into the lower hole using construction grade adhesive. Cross holes would be drilled though the sides of the upper post to accept bolts that would be threaded into prepared holes on the other end of the steel pipe. See picture below:
Best Answer
There is a product called an 'Ezy-out' (or some variation) and other brands of similar products which are basically a reverse thread, tapered, coarse pitch tap or drill-bit. You can use these by hand or in a drill - in reverse-, once it bites into the screw it will begin driving itself in further or unscrewing the stuck screw.
What size and material is the stuck screw?
Liverally coating the screw in a penetrating lubricant like WD-40, applying heat to the screw to make it expand or tapping it with a hammer while trying to remove it will all help to some dregree.