No .. Sub luminal speeds are no problem, and the speed at which light waves propagate in vacuum, is independent both of the motion of the wave source and of the inertial frame of reference of the observer (which has been abundantly verified via measurements).
IOW the speed of light in a vacuum, is a universal
physical constant.
I am unclear what that has to do with your above question??
'Doppler shift' is not
an explanation for the concluded age of the universe(??)
Expansion is inferred from the observed recession of distant objects.
I can't see how 'the rate of expansion' somehow acts to cancel out your words(?) I mean,
the rate also appears to have changed over the period of time to the present.
I do not have a clue what you're talking about here .. you seem to be very confused on this topic(?)
I don't understand where you're coming from on this.
Others may care to explain further but until you get the basics clearly nailed in your mind, you should refrain from making accusations about others' understanding and how they have arrived at it.