This belongs in intro physics...
The only thing wrong with your solution is your E4 - E1 calculation. Think about what it means to go from n=1 to n=4 and the energy DIFFERENCE between these 2 levels (Hint, your answer by definition cannot be greater than 13.6eV)
If you know the total time a projectile is in the air, you can figure out how long it took to get to the top of it's trajectory (it's highest point) by just dividing by 2. [The projectile will take the same amount of time going up as down]. Then just solve for v_0 by using kinematic equations.
P_air is the pressure of the air INSIDE the cylinder. you basically have 2 forces acting on the plate, up and down. The "up" force is atmospheric pressure that you can calculate knowing the pressure and the surface area. The "down" force is actually the sum of 3 forces. The weight of the plate...
Think about it a little harder, you have the general equation right but there are some coefficients and simplifications you can make. (Specifically a factor of 1/2 in front of the PE? and D=x+d, and d is what you want.)
You know the magnetic fields go either clockwise or counterclockwise around the wire. To find the direction, imagine the thumb of your right hand points in the direction of the current, then your fingers will curl in only one direction and that is the direction of the magnetic field.
And btw...
Try to draw it on a piece of paper...you will see that because the infinite line is along the x-axis, the distance from the line aka the distance from the x-axis doesn't depend on x at all. Only y and z. Then you can see that if you move up 1 in the y direction and 1 in the z direction you have...
I can't see the graph very well...but slope is \frac{\Delta\;y}{\Delta\;x} so as long as the y-axis is Force (in N) and the x-axis is displacement (in m), then the slope would be N/m or k.