According to Einstein's assumption, light always travels at 'c'. But, according to him, only the relative velocity changes. Isn't this a wrong assumption? Because, everything has to gradually lose energy. Won't photons dissipate an infinitely small amount of energy as they move, making them lose some energy?

Second part of the question: Will gravity affect the movement of photons? Because, if they have zero mass, only then there's no gravitational effect. This is due to considering their 'rest' mass as zero. Shouldn't it just be negligibly small?

Photons will not slow down. If by some mechanism they lost energy then the wavelength and frequency would change, but the speed would remain the same.

 

Gravity affects light; this is predicted by General Relativity, in which gravity is a geometric effect that curves space, and light follows a null geodesic.

So, as the frequency changes, the color of the light should change. Does this happen?

When lasers are bounced back from distant objects, if the frequency etc change, shouldn't the information being gathered have disruptions?

So, as the frequency changes, the color of the light should change. Does this happen?

When lasers are bounced back from distant objects, if the frequency etc change, shouldn't the information being gathered have disruptions?

 

Yes, the color changes. Whether information is disrupted, it would depend on how the information was encoded. The shift sometimes provides information, e.g. the redshift of a spectral line (or laser line) tells you how fast the object is moving along your line-of-sight, via the Doppler shift.