Basically you have raised questions those I have listed below.
(1) Question in relation between the energy of light and it's frequency.
(2) Question in light loosing energy and so decreasing it's frequency as it escape a gravitational well.
(3) Question the test in 1962 showing time running slower at the bottom of a tower near Earth than the top of the tower where time is running faster.
Accordingly, I would like to answer your questions as mentioned below.
A1. Photon is gauge boson, carrier of electromagnetic force. Therefore, the entire spectrum of electromagnetic waves carried by photons.
Light is a small part within the spectrum of electromagnetic waves, the spectrum includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.
The energy of a photon can be calculated from Max Planck's equation E = hf, where, E denotes energy of a photon in Jules, f denotes frequency of the wave in Hz, and Planck's constant h = 6.625×10^–34 Js. The equation is applicable for the electromagnetic spectrum.
Here is the relationship between frequency of photon and it's energy.
A2. When a photon leaves a gravitational well after its emission, it spends energy to escape the gravitational well, as a result it's energy reduces, and as per the Planck's equation above, photon frequency too reduces due to the reduction of it's energy. The photon will have most of it's energy immediately after it's emission for it's source and so it's frequency too would be most then.
The wavelength of a wave is inversely proportional to the waves frequency, so that when a photon reduces it's energy, it's frequency too reduces but it's wavelength increases, resulting the red shift of the photon in electromagnetic spectrum, due to such enlargement in the photon wavelength. This is known as gravitational red shift.
Light consist of photons, so when photon energy reduces, it's frequency also reduces, as explained above.
A3. The test in 1962, that you have questioned, showing time running slower or faster depending upon the relative gravitational potential differences.
In fact, it is an erroneous proposition in relativity. Because of the relative slower or faster running of the clocks are not due to time dilation, rather for the error in the reading of relative clock times due to relative phase shifts in frequency of the wave of the clock oscillations under relative gravitational influences, and corresponding relative reduction or enlargement in the wavelengths of the wave of said clock oscillations, in a relationship between wavelength and time period of the waves of the clock oscillations. Distortions of wavelengths exactly correspond to time distortions λ∝T.
Experiments made in electronic laboratories on piezoelectric crystal oscillators show that the wave corresponds to time shift due to relativistic effects.
Whereas, the time interval T(deg) for 1° of phase is inversely proportional to the frequency (f). We get a wave corresponding to the time shift.
For example, 1° phase shift on a 5 MHz wave corresponds to a time shift of 555 picoseconds (ps).
We know, 1° phase shift = T/360.
1° phase shift = T/360 = (1/f)/360.
For a wave of frequency f = 5 MHz, we get the phase shift (in degree°)
= 555 ps.
Therefore, for 1° phase shift for a wave having wavelength λ = 59.95m, and frequency f = 5 MHz, the time shift (time delay) Δt = 555 ps (approx).
Time shift of the caesium-133 atomic clock in the GPS satellite in space:
For 1455.50003025° phase shift (or, 4.043055639583333 cycles) of a 9192631770 Hz wave; time shifts (time delays) Δt = 0.0000004398148148148148 ms (approx) or, 38 microseconds time is taken per day.
Therefore, the wavelength dilation of the clock oscillation due to relativistic effects, or gravitational potential difference on the clock mechanism results in corresponding error in the reading of time in the clock, wrongfully presented as time dilation. Time dilation is rather wavelength dilation.