How to interpret the velocity from relativistic doppler effect/equation for the redshift of cosmic background radiation? Value and calculation of redshift of the microwave background radiation:
https://lambda.gsfc.nasa.gov/education/graphic_history/microwaves.html
https://thecuriousastronomer.wordpress.com/2015/07/30/what-is-the-redshift-of-the-cosmic-microwave-background-cmb/
We substitute it into the Doppler formula for light and determine the speed from it.
What and in relation to what is this speed? From this speed, time dilation and length contraction can be determined. In relation to what?
In ordinary Doppler, we distinguish between the speed of the source and the receiver. In the photon Doppler we do not have this distinction, because the source and the receiver can have a velocity defined only relative to each other, so for the source and the receiver it is the same, only in the opposite direction.
What could be both a source and a receiver at a redshift of factor 1100?
In our reference frame - Is there a redshift of light, that cannot reach us?
 A: It is the velocity of Earth relative to a huge spherical shell of what was plasma back when the CMB was emitted. That shell was just part of a roughly uniform plasma which was, on average, not moving relative to the  average distribution of matter in the universe; that average distribution picks out one reference frame among all the others. It is called the co-moving frame. The reason one can pick out such a frame is that the matter is distributed with close to homogeneous density on the largest scales and close to homogeneous velocity distribution also (i.e. isotropic with regard to any chosen centre). There is evidence however of small inhomogeneities even at very large distance scales and this impacts on some of the interpretation of evidence of the dynamics of the expansion.
For the second part of the question, regarding the frequency shift, the notion of a relative speed of source and detector is not needed. Rather, what you want is the calculation of momentum conservation along a null geodesic. The momentum conservation follows from the field equation when the metric has a certain form. The light follows a null geodesic. The frequency shift is the difference between what is observed by observers fixed relative to the co-moving frame at different spacetime locations. Another way to think of it is to think of two null geodesics following the worldllines of successive wavefronts. These two geodesics draw apart along with, and indeed in exact proportion to, the universal expansion.
A: The Special Relativistic Doppler effect formula should not be used to interpret cosmological redshifts. Mandatory reading is section 3.1 of Davis & Lineweaver (2003).
Cosmological redshifts are due to the emission of a photon in one frame of reference and the receipt of that photon in an entirely different frame of reference at a later cosmic epoch. They are caused by the expansion of space, not the movement through space. They can be interpreted in terms of recession velocities but only approximately obey the Special Relativistic formula at low redshifts
In terms of a "recession velocity" (rate of change of proper distance), the surface of last scattering - from where the cosmic microwave background is released - is approximately 3.2 times the speed of light; the exact value depending on the adopted cosmological parameters. Clearly this value wouldn't be possible using the special relativistic formula.
Plot below from wikipedia: The redshift of the CMB is about 1100.

If you do choose to interpret the velocity from the Doppler shift formula, then that is the relative velocity between the gas that emitted the radiation then to us now (see Bunn & Hogg 2009). However, this is not the velocity given by Hubble's law, which relates rate of change of proper distance to the proper distance - i.e. the separation of two objects at the same epoch.
In your edited question you ask a further question - the measured redshift could be arbitrarily high. In practice we will not observe light with redshifts higher than 1100 because the universe was opaque to radiation prior to the epoch when that was emitted. However, in principle we might measure gravitational waves with higher redshifts from the earlier universe.
A: The speed we are talking about is the space expansion rate itself.
There is no redshift of light, that cannot reach us because of the expansion with v>c.
These are conclusions, we can draw from it:

*

*The velocity determined by the Doppler formula after substituting the redshift of the microwave background is $0.9999983471 ~ c$.

*The speed determined by the Hubble formula after substituting its current value and the distance equal to the age of the universe $\cdot\ c$ is $1.03 ~ c$. I guess the total error is $3 \%$, which is exactly the difference.

I check, that the Hubble constant and the age of the universe are not determined from my background radiation shift and from it from the speed obtained from the Doppler:

*

*However, obtaining a true value for Ho is very complicated. Astronomers need two measurements. First, spectroscopic observations reveal the galaxy's redshift, indicating its radial velocity. The second measurement, the most difficult value to determine, is the galaxy's precise distance from earth. Reliable "distance indicators," such as variable stars and supernovae, must be found in galaxies. The value of Ho itself must be cautiously derived from a sample of galaxies that are far enough away that motions due to local gravitational influences are negligibly small.


*Measurements by the WMAP satellite can help determine the age of the universe. The detailed structure of the cosmic microwave background fluctuations depends on the current density of the universe, the composition of the universe and its expansion rate. As of 2013, WMAP determined these parameters with an accuracy of better than than $1.5 \%$. In turn, knowing the composition with this precision, we can estimate the age of the universe to about $0.4 \%$: $13.77 \pm 0.059$ billion years!
How does WMAP data enable us to determine the age of the universe is $13.77$ billion years, with an uncertainty of only $0.4 \%$? The key to this is that by knowing the composition of matter and energy density in the universe, we can use Einstein's General Relativity to compute how fast the universe has been expanding in the past. With that information, we can turn the clock back and determine when the universe had "zero" size, according to Einstein. The time between then and now is the age of the universe.
This is from the site and from the nasa document:
https://wmap.gsfc.nasa.gov/universe/uni_age.html
https://btc.montana.edu/ceres/html/universe/hnought.htm
Would you agree that these are two different methods, one of which is wildly experimental and the other is brutally bold? Can you fault the second method, and if not, then you have to ask yourself what can now be calculated exactly, knowing this value to the $6$th decimal place. The error of the background radiation shift itself, which I did not dig up, so to speak, almost disappears even at $5 \%$ due to the calculation $v = c \frac{1100^{2} - 1}{1100^{2} + 1}$ giving the difference from $1$ in the sixth decimal place.
In the package you get a universal frame of reference - the CMBR frame in which the background photons from all sides, passing through its origin, have the same wavelength. Another thing is that there are infinitely many such systems and they all move away from each other, so it will probably be hard to use it.
Consequently, we have a dependence of CMB temperature on the expansion speed and on time, that is the age of the universe.

*

*An almost perfect black-body spectrum is exhibited by the cosmic microwave background radiation. (...) Relativistic Doppler effect: This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of $c$. An example is found in the cosmic microwave background radiation (...)
https://en.wikipedia.org/wiki/Black-body_radiation#Doppler_effect


*Temperature will drop to zero, when the expansion rate will reach $c$, not the other way around. Logarithmic plot of temperature in function of time:
https://i.stack.imgur.com/SwyJJ.jpg
You can extend it to the $x -$intercept.
Either light will stop reaching us, starting from the most distant galaxies, or spacetime will tear itself apart.
Imagine a photon stretched in space along the direction of its motion from end to end of photon's world in range equal to the age of universe $\cdot c$. In this way, each photon of background radiation is stretched, and such a photon passes through every point of space from each side. And what will happen to these photons, when the universe in their direction of motion, according to Hubble's law, stretches more than it is allowed by the limit of speed in Doppler formula, which must confirm their redshift? It's going to be the square root of the negative. There is no redshift of light, that cannot reach us because of the expansion speed $v>c$, but the velocity in Doppler equation that gives the redshift of CMB photons that are everywhere, this velocity doesn't know it and it's going to exceed $c$, resulting in a square root of negative and previously zero in the denominator for $v=c$.
So I have a prediction of when and how our universe will end, and now I'm guessing what will come after it. Universe will be switched over to the other side. Spacetime will tear apart and sew together on the other side of the event horizon of the black hole we'll find ourselves in. It will tear, cross the horizon, sew together, and the expansion will change direction with respect to $c$, which means a big-bang from the inner surface of the event horizon of the black hole, where we will be, towards its center, and for us it will probably be like a new, normal universe, or on the contrary: a universe in which time will be swapped with space in equations and in reality.
Such an inverted big-bang from the inner surface of the event horizon would explain this:
https://www.esa.int/var/esa/storage/images/esa_multimedia/images/2013/03/planck_cmb/12583930-4-eng-GB/Planck_CMB_pillars.jpg
As a result, we have one big, cosmic oscillator. We have successive, space-time swapping universes, that are alternately holes and their complement, yin and yang.
$t_v=13.77\pm0.059\ \cdot 10^9\ y$
$v=0.9999983 \pm 0.00000??\ c$
$v = H_0 \cdot d_v = H_0 \cdot c t_v$
$H_0 = \frac{1}{t_v} \frac{v}{c}$
$c = H_0 \cdot d_c = H_0 \cdot ct_c$
$t_c= \frac{1}{H_0} = t_v \frac{c}{v}$
$\Delta t = t_c - t_v = (\frac{c}{v} - 1)\ t_v =$ ;)
I'm explaining this calculation only because moderator demanded it. It's a calculation of the time left to the end of the world. I wanted you to figure it out yourself.
Explanation of CMB photon travel through the expanding space
Space, that has been traversed by CMB photon does no longer exist, as well as the space that has been expanding behind it, moving away its emission place, as well as the space that has been expanding in front of it, lengthening its path. Current space, that separates us from the current position of the place of its emission, this space has a current metric, other than the metric of space traversed by the photon, other than the space that was expanding in front of it and behind it during its travel. In this current metric, our distance to the place of CMB photon emission is equal to the universe age * c. Former space no longer exists not only because of the changes of matter density in the expanding space traversed by the photon, but above all because the spacetime exists only for the time of its metric. What we have is current spacetime, photon travel time and its velocity. There's only one way to calculate the distance from this data.
Thought experiment with an ant walking on stretched rubber band
I lay the long rubber band loose on the table and stretch the short one. Both have ants starting simultaneously from one end to the other, moving relative to the fabric of the rubber band with v=c. I stop stretching the short rubber band when it reaches the length of the long one, and exactly at that moment both ants reach the finish line. In both cases, they've traveled the ct distance. There is a catch. A short rubber band has a greater linear density (closer scale) at the start and it is in relation to this density/scale that the ant moves with v=c. After stretching the rubber band, its density decreases to the density of a long rubber band, their graduations coincide/overlap. And since the motion with v=c is relative to the density/scale, the varying rate of stretching loses its meaning because both ants always come together, so to speak, as long as I stretched the rubber band at a rate less than c, the speed of an ant. Otherwise, the rubber band breaks and the ant does not come. And that could even be nicely animated.
The scale coincides with the redshift or the wavelength of the CMB photon - it comes to the same thing.
That's why the changing rate of expansion - as long as its speed remains less than c - has no effect on the size of our visible universe. Its radius is always ct and that's why wikipedia is horribly wrong and probably half the astronomy with it.
Another idea of how CMB photon travels through the expanding space
In this vision, CMB photon is literally embedded in a single, expanding cell. You could say that its motion is apparent. At the same time it always travels with v=c relative to the neighboring cells, but in a static, snapshot frame, like the ones in the drawing.

