I would have the poster mounted to a substrate before attaching it to the frame. foamcore is a good choice if the poster is not valuable. Acid free materials are worth the extra money if they are in contact with the poster. If you mount the paper poster to the wood frame it will never look very good. It will show the frame elements through the paper and the glue will cause differential shrinkage and buckling. If this is just a digital print that could be replaced this is what I would do; mount the poster to foamcore with spray adhesive taking care to avoid wrinkles. Mount the foamcore to your wood frame with double stick foam tape around the entire frame including cross pieces. This will keep the poster from showing the frame members though the paper and will Make an attractive presentation for a year or two. If you value the poster as an artwork, it is irreplaceable, have it done by a professional framer with good materials.
That's a hard thing to google. I can offer 2 possible approaches:
1.) I think you could do something like these doors with battens:
You would have to turn the battens 90 degrees from the drawing since your facing boards are horizontal. The most important part would be the vertical battens on the left and right side. This would effectively form a frame with the top and bottom boards.
2.) If you want the frame to be truly internal, you could build a mortise-and-tenoned rectangular frame, and then apply facing boards to both sides. For more info on this, I would google "frame and panel doors". You could do the simple rectangle, like cabinet doors, and leave out the panel. This frame then could be sandwhiched between layers of facing boards. If you don't feel like doing mortise and tenons, substitute loose tenons, biscuits, dowels joints, pocket screws, etc.
Either way, screws or nails should be fine for attaching the facing boards to the frame. If you use nails, hammer them in at slightly varying angles to prevent the boards from pulling straight out.
Also, leave a little space between the facing boards. Here in the US, I would use a dime as a spacer, but you could use a couple business cards, or a small drill bit, or measure a millimeter, etc. This allows the wood to expand and contract with temperature and humidity changes.
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Easiest way is to cut one in half, but I'll assume you want to do this non-destructively.
One way that you can check is to remove one of the hinges. If it's a recessed hinge you'll find yourself looking at the inside of the door. If it isn't recessed you can examine the screw holes and/or probe them with a dental pick.