What is the whole picture of the NP-hardness proof of Super Mario Bros?
Graph Connection and Gadget Orientation

Short answer to your first question is to use graph algorithms to compute gadget assembly. Erik mentions this briefly. Keep in mind the gadgets don't necessarily have to be oriented as shown above. You could rearrange the gadgets however you wish as long as you retain the edge connections and entrance/exits of the graph. 

The idea of "falling down" from a variable is so that there is no way to get back up to it. After you fall down there could be a tunneling system or corridor that allows Mario to get to the bottom side of a clause and hit the question box from below. 

Right-to-Left Check Path

You could use the mirror (about the y-axis) of the clause gadget to achieve the capability of going right to left. This would be true for any of these gadgets as long as you kept the entrances and exits hooked up properly using tunnels. You do not gain or lose anything from going right to left versus left to right in SMB. They would be equal movements and since the whole level is on the screen at one time, we are not restricted from going right to left. 

The Crossover Gadget

The crossover gadget is placed where edges intersect (edges between literal and clause). This is because SMB is a 2-dimensional game and thus the graph 
using all these gadgets must be planar. The crossover gadget will then act as a vertex where the two intersecting edges can safely connect and not interfere with each other. They will not interfere because if you enter top left, you can only leave top right and if you enter bottom left you can only leave top center. More on the planarity and edge interference here.

Importantly, the crossover gadget is directional. You can only go left to right through the gadget once. This means the edges must also be directional. Whenever you see an edge from literal to clause there are really two directional edges; (literal -> clause) and (clause -> literal). Which results in there being four crossover gadgets whenever the two directional edges cross with two other directional edges. Erik explains more in the video at 1h1m20s (It won't allow me to link to it because this is my first post).