Relativity theory - lower end of falling to Earth object moves slower or faster than higher end? According to relativity, time slows close to gravitational field (I prefer to say the processes with which we measure Time slows).  (https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment) - "General relativity predicts an additional effect, in which an increase in gravitational potential due to altitude speeds the clocks up. That is, clocks at higher altitude tick faster than clocks on Earth's surface. "
Thought experiment:
Earth and Object. Earth is significantly more massive then Object, we can neglect gravitational relativistic effect of Object mass.
Earth is in inertial movement and there are no other significant mass expect Earth and Object. 
Clocks are fixed relative to Earth, so simultaneity of measurements can be assumed (correct me if not).
Object falls down to Earth. Lower part is closer to Earth center so hypothetical clock in its' location is ticking slower than clock located in space where upper part is. Could we say lower part falls down faster than upper then to make object falls down as whole at same speed to resting to Earth observer? Or object is distorted to compensate for clock speed difference as hypothesized below?
Tidal force should stretch the object, attracting lower end stronger. Maybe tidal exactly cancels out difference in time frames? If yes, can it be shown by formulas?
Or not maybe time effect can be explained by that lower part of object 'experience' quicker length contraction as it falls than upper (https://en.wikipedia.org/wiki/Length_contraction), although it's said contraction due to speed only in wiki, gravitation not mentioned? 
Or by relativity of how we can measure space (don't know if there is such term/topic)? Or other explanation?
 A: Take 1
To see clearer, let's simplify. Your system is completely equivalent to 2 observers (posts of a ball) moving in an accelerating frame (falling down). This is basically the setup of Bell's spaceship paradox and you should read about that. We will make an assumption of homogeneous gravitational field, so the acceleration is constant - but we have to do this to avoid math.
What is effect of acceleration in them? There is no effect, they fall freely and do not experience anything. But if you would compare the time on the clock on the ground and those in observer's hands - you will see that observers experienced less "proper time". But time between observers since some height will be exactly the same.
You mistake the difference in proper time between reference frames as having effect on the velocity. This is incorrect, from ground's perspective the infall speed will be exactly as you expect classically from a falling body.
The distance between observers will indeed change, causing strain on the ball, but you shouldn't try to compensate it by "tidal forces" - if you consider general relativity, it replaces all gravitational phenomena entirely.
Take 2
(after update)
Okay, let's take back the homogeneous gravitational field. Again, it is better not to consider an object, but 2 points — even in classical physics object falling in the inhomogeneous gravitational field is strained and deformed. I definitely do not want to deal with that, because strain will cause counteracting acceleration and the whole picture blurs.
Let gravitational acceleration linearly increase. Now, split the time of fall into small ranges where acceleration increases negligible. In each slice the previous picture is valid, but with ever increasing acceleration.The time measured by observer's clock will be even smaller.
My quick numerical simulations confirm this picture, but at this level of discussion GR notation and formulas will only add confusion.


Clocks are fixed relative to Earth, so simultaneity of measurements can be assumed (correct me if not).

Simultaneity is possible only if 2 events happen at the same point at the same time. Otherwise it is impossible to know of the other event.

Could we say lower part falls down faster than upper then to make object falls down as whole at same speed to resting to Earth observer? Or object is distorted to compensate for clock speed difference as hypothesized below?

Object is distorted in the varying gravitational field even without any relativistic effects.

Tidal force should stretch the object, attracting lower end stronger. Maybe tidal exactly cancels out difference in time frames? If yes, can it be shown by formulas?

You want to double-count the gravitation — once as GR, once as classical. In general, once you start talking about GR — forget about gravity. In GR everything is explained by a curvature of space. I know just the video that explains this in layman's terms: https://www.youtube.com/watch?v=DdC0QN6f3G4

Or not maybe time effect can be explained by that lower part of object 'experience' quicker length contraction as it falls than upper (https://en.wikipedia.org/wiki/Length_contraction), although it's said contraction due to speed only in wiki, gravitation not mentioned?

Parts of the object do not experience any length contraction or time dilation. If something is not moving — there are no relativistic effects. These effects appear only on objects moving with respect to the observer.

In general, I do not see any paradox here. If you do, please cleanup the question from wrong statements that I tried to explain and we can continue.
Thank you for the reference to a cool experiment — I did not know about it!
