Minimum speed required to clear rectangular object I believe this comes under the "homework question" category, I will try to make it as general as possible to avoid concerns about cheating/extra help.
The problem is to find the minimum initial speed required to launch a projectile over rectangular object of width $x$ and height $y$.
My question is : is it a correct assumption to state that the tangent to the path of the velocity at the point where it passes the first corner of the rectangle is at 45° to the horizontal?
 A: 
My question is; is it a correct assumption to state that the tangent to the path of the velocity at the point where it passes the first corner of the rectangle is at 45° to the horizontal?

You're somewhat conflating things, as the figure below should make self-evident:

The optimum launch angle $\theta$ is 45° ($\frac{\pi}{4}$) but that's not the angle at which the projectile will 'clear' the first left hand corner.
At $\frac{\pi}{4}$ it becomes a matter of finding the intial velocity $v_0$ and the distance between the launchpoint and the mid point of the base of the object.
This Wiki entry should make calculation of these parameters quite easy.
A: You are correct. The minimum speed $v$ needed to traverse the roof requires an angle of $45^{\circ}$ at each side of the roof. Applying the formula for maximum range equal to the width of the roof will give you the speed $v$ at each edge of the roof as a function of $x$ and $g$.
The speed $u$ required on launch from the ground is then easily found from conservation of energy : the KE lost in getting up to the edge of the roof equals the PE gained. The result is a very simple relation between $u,x,y$ and $g$. 
