It looks like a nice piece of work! :)
What is the purpose if I may ask?
Since you asked me to critique your work, I'll try to suggest a couple of improvements.
Newton's third law states: action = -reaction.
Where the paddler and the boat interact, this law must hold.
In particular this means that the force of the seat on the paddler must be equal and opposite to the force of the paddler on the seat.
What I mean, is that in your equation (2) en (3) ##F_{seat}## should have opposite signs.
Same thing for ##F_{footbar}##.
In equation (3) this is positive, but in equation (2) it should be negative.
You have ##F_{dragAir}##.
I'm assuming that is the drag of the air on the paddler.
But this force does not act on the boat, but only on the paddler.
In your kayak stroke (equation 5), you should see only the so called "external" forces.
The boat, paddler, and paddle together form a system.
The forces on the footbar and the seat are not external forces.
They (should) cancel when you add equations (2) en (3) together.
This would leave you with only ##F_{paddle}## and ##D##, which are the external forces.
Good luck!