In picture #1, that is a thrust plate, often original, sometimes added on, or beefed up. It is like a 'scuff pad', and adds protection and reinforcement. The thrust of the motor pushes on that sheet, also, think of the top of the transom as a 'pivot point' when the motor is mounted, this plate again, adds protection to the skin of the transom.
To summarize, in Pic #1, the plate you mention is the 'motor mount', which is to protect the transom skin.
In Pic #2, Yes, I would put the wood right up to the lip, if you want it extra thick, I would bond two pieces of wood together, personally, I would make the transom wood the whole width and height of the boat, if weight allows.
Also, the vertical piece running from the floor of the boat to the tansom, is a 'knee', it supports the transom from flexing from the weight and thrust of the motor. Holds the transom rigid. If it is easy to replace, you can install two knees, like Mirrocraft does.