Why do we launch rockets during the day? The clips that I have seen of rockets launching all seem to be carried out during daytime. However, we learnt at school that rockets are fired closer to the equator and towards the east to take maximum advantage of Earth's rotational motion, getting a boost in speed of roughly $460m/s$ at the equator. If we would go to such lengths to gain a boost of speed of $460m/s$, why don't we take advantage of the fact that the earth orbits the sun at the massive speed of $30km/s$ to get an extra boost in velocity relative to other stellar objects? But to take advantage of both the rotational speed of the earth about its own axis and the rotational speed of the earth about the sun the rocket would have to be launched at nighttime:

as the only times when the boost from the purple rotation was in line with the boost from the red rotation is during the night-time. It seems that this would greatly reduce the fuel needed to reach distant objects (for instance Pluto: New Horizons seems to have launched during the daytime). 
 A: Speculatively building on the answer given by @fibonatic here:
If you were instantly "far from earth", then your argument for launching at night makes sense for long distance travel. But if you consider that you are orbiting Earth for a short while as you gain altitude, you actually "sling shot" around the earth by starting on the day side - the gravitational pull from the earth accelerates you in the direction you ultimately want to go, and you actually end up overtaking the earth:

This means that the correct place to launch will be determined not just by your instantaneous velocity at the point of lift-off, but also by the rate of climb that you can sustain.
A: For missions beyond Earth orbit, it's very common to launch into a parking orbit around Earth before translunar or interplanetary injection.
Therefore, the time of launch doesn't constrain the departure angle from Earth; you can get any departure angle you want by waiting no more than 90 minutes. This allows more flexibility in launch time even if the window for the injection burn is very short. 
That said, night launches aren't that uncommon for e.g. direct launches to geosynchronous orbit, although daytime launches seem generally preferred for practical, ground-logistics reasons.
A: When launching into a low Earth orbit only your velocity relative to the Earth matters, as seen from the not-rotating reference frame of the Earth. Your velocity relative to the sun does not matter, because once you are in the orbit your velocity vector relative to the Earth will oscillate between pointing towards and away from the velocity vector of the Earth relative to the sun.
When performing an interplanetary transfer the Earth's velocity does matter. Usually such transfer is performed when in low Earth orbit. So if you want to travel to space outside Earth's orbit, then you want to leave Earth's "gravity" in the same direction as its velocity relative to the sun, also called prograde. But because the Earth will also slightly curve your escape trajectory you will have to burn while near trailing side of the Earth (where the sun is setting) such that you pass behind Earth's night side. The opposite is true when you want to go to space inside Earth's orbit.
