Gantry sides:
2 spaced apart 5mm plates will be stiffer than a single 10mm plate, but they need to be well connected to each other to make it work, otherwise you are wasting all the effort in creating it. In a simple RHS section they are well connected to each other by the other 2 sides so they can transfer the load from one side to the other and give the stiffness benefit.
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In your design they are only connected periodically with spacer posts so it will not be a stiff as a RHS section the same overall outer dimensions as your spaced out design, although it will be better than a single 10mm plate.
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Are these steel plates? If aluminium I think they would not be stiff enough. You could add a layer of plywood bonded in between (as well as the spacers) to try and transfer the load better, plus maybe add a bit of damping. Better still would be just use 20mm aluminium plates as this is simple and is proven to give reasonable results on a raised gantry router.
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Gantry beam:
Do not use 5mm flat plate for the beam - it will be very poor for stiffness. You might be thinking that the rails would add a bit of stiffness, and they will do a bit of that but the overall performance will be poor as they can twist easily about their own longest axis.
The 40x20x3 steel profiles are much better than plate, although depending on what you want to cut that is still on the small side. Much better and suitable for cutting more materials would be 50x50 as a minimum.