Is the speed of light constant

[Dale]

The Earth is moving through space at a minimum of 7500kps.

"Moving through space" is a nonsense phrase. The Earth is moving at 0 in its rest frame and at 7500 in some other frame (e.g. the rest frame of the local CMB radiation). There is no "rest frame of space".

 

[John15]

If the Earth can have a rest frame of 0 whilst moving through space then that must apply to all including wavelengths of energy, everything is governed by the same rules.
Its obviously not the Earth that has the rest frame though its us as the observers having a 0 rest frame relative to the earth, we cannot know however the Earth's relative speed to anything outside the solar system with any accuracy.
I am guessing however that c is acctually a universal frame of reference as it is the only thing that has constant uniform speed to which all other speeds can be related. Its actual speed is therefore unimportant just that its uniform.

 

[HallsofIvy]

Light obviously reaches the speed of light.

That is not only not "obvious", it is untrue. Light does not "reach" the speed of light is at c from the instant it is created. Light always travels at "c".
(Saying "light always travels at the speed of light" is subject to misinterpretation!)

Question is then what happens to it at this speed that stops it going any faster?

 

[Dale]

If the Earth can have a rest frame of 0 whilst moving through space then that must apply to all including wavelengths of energy, everything is governed by the same rules.

Yes, everything is governed by the same rules and those rules say that something traveling at c in one inertial frame is traveling at c in all frames. This implies that it is not traveling at 0 in any inertial frame.