Relativity gedanken experiment Lets consider the following thought experiment: A spaceship is in circular orbit around Earth traveling at 99% of $c$ (the orbital distance is chosen in such a way that inside the ship there are no centrifugal force effects). Aboard the ship there is an astronaut connected with the space center flight room on Earth through 2-way video cameras and monitors system in such a way that the scientists from the flight room can see the astronaut in the ship and also they can see themselves in the astronaut's video monitor on the ship. Similarly the astronaut can see the scientists in the control room and himself on the video monitor situated in the flight room. 
The question is: what will the astronaut and scientists see on their respective monitors? From my understanding of SR the people on earth due to the time dilation effect on the ship should see on their monitor the astronaut almost "frozen" in time since he's moving at 99% of $c$, while the astronaut should see on his monitor the scientists on Earth moving extremely fast similarly to a fast forwarding movie. Is this correct? If yes, then here is the most interesting part of this thought experiment: what will the scientists on Earth see when they are looking at the astronaut's monitor inside the ship and see THEMSELVES in it? Will they see themselves the same way the astronaut sees them (moving very fast) since the monitor is inside the ship or will they see themselves at the "normal" familiar speed at which they live? Apparently there is a paradox here so which is the correct answer and why? 
 A: Assuming you fire rockets to orbit the earth at a high speed, then indeed the orbiter sees the images of the flight room clock going faster than their own. And the flight room sees images of the orbiter clock going slower than there own.
The scientists in the flight room see images of themselves moving at normal speed, but from the past (however long it took the signal to get to the orbiter and back).
I saw no paradox and see nothing to explain. Plus, it is a misconception to think you need high speeds to have relativistic effects. You can measure relativistic effects at walking speeds, and people have. You just need sensitive detectors. But then you need to consider general relativistic effects too. The point of the high speed was to get the special relativistic effects to dominate.

"what will the scientists on Earth see when they are looking at the astronaut's monitor INSIDE the ship and see THEMSELVES in it?

They see the images of themselves moving faster than the images of the orbiter, i.e. moving at regular speed.
Indeed if the orbit moves at a gamma factor of 5 then they see themselves at normal speed, which is five times as fast as the orbiters clock. And the orbit also sees themselves move five times as slow as the images from the flight center.
Everyone agrees that the image of the flight center clock ticks five times faster than the image of the clock on the orbiter. The person on the orbiter sees this with their own eyeballs. The people on the flight center see it.

Will they see themselves the same way the astronaut sees them (moving very fast)

No one sees fast or slow. They sees faster or slower. Everyone sees the the images of the flight room play at a faster rate than the orbiter. The flight center just sees themselves at normal speed. You never even made a case for why they would see anything else.
The orbiter just ages slower.

So this is the part that needs explaining.

What is there to explain? The orbiter makes a path in spacetime, that path has a certain amount of proper time per orbit. And more proper elapses for the flight center. Each person's clock ticks based on the proper time of the path the clock makes. The proper time of the orbiter is less than the proper time of the flight room.
If the flight room clock ticks once a second then the orbiting clock gets five images of ticks from the flight room for each it makes itself. And it sends all six to the flight room.
What does the flight room see? It sees one of their own ticks come back from the orbiter each second and once every five seconds it sees a tick from the 
Come on, you seeing a slow person watch a normal speed movie is what it looks like when you see a movie watched by a slow person that claims the movie is running fast.
You can also draw the spacetime diagram. Mark five places along the circle as places where the orbiter ticks. Mark twenty five times the flight room sends out an image. Notice that the orbit gets five for every one, so they see the movie from the flight room playing at five times speed. And the flight room sees each of the twenty five ticks sent back at a regular speed with only five ticks of the orbiter.
They see the orbiter going slowly watching a regular speed movie and they know the orbiter would claim the movie is going faster than them. Everyone agrees the orbiter sees five flight room ticks for each of their own.

The second part of your comment is of no relevance since of course its true that relativistic effects happen at any speed with the corresponding "dominance" effect.

I had to bring up rockets and acceleration since otherwise your question was contradictory. If you agree that an accelerating orbiter is what you want then you can read my answer.

The reason I have chosen such an extreme speed was to make an easier and clearer mental image since the crucial effect in this experiment is time dilation.

It makes people confused about what you are trying to do since things don't naturally orbit the earth at such high speeds.

I hope that you can see now why there is a paradox.

No. I never saw a paradox. I learned how to compute the proper time of curves. And I know that second hands of clocks tick when the proper time of the curve the clock travels is 1 second. And this doesn't cause paradoxes. It allows me to know when clocks tick when they move on paths in spacetime. Knowing doesn't cause paradoxes.

You said that A will see on his monitor E moving as in a fast forwarding movie.

Yes. That's what happens. But there is a similar situation that happens without Special Relativity, so maybe you should step back and consider this simpler situation. A man has two twins as children and gifts them with a watch each. But one of them runs slower than the other. The one A with the slower running watch says to their twin, I see your watch E tick at five times the rate mine A does. The other twin says they see their own watch A tick at the same rate as their own watch A and sees watch E tick at 1/5 the rate. Furthermore they see E seeing watch A tick at 5 times the rate as E.
No Relativity. And no paradox.

If this is the case then E back on Earth will see exactly the same thing since the camera is on the head of A. 

Yes. They see the clock on A ticking at 1/5 the rate as the clock on E. Everyone sees that. It's what happens.

But how can that be since they are not moving as in a fast forwarding movie back on Earth? 

No one (other than maybe you) ever claimed that they would see themselves moving fast. There is no fast. Just faster.
When I say that A sees the movie from E playing like a fast movie, I mean that they see the clocks in the movie tick at five times the rate they see their own clocks tick. I don't mean anything magical like that they can tell if movies are fast, only that they can tell they are faster than their own clocks.
Imagine you watching a time lapse movie where things appear to move fast. And imagine someone recording you watching that time lapse movie. Now imagine they play that recording of you watching the movie but they play the recording in slow motion. There isn't a paradox there either.

Since the eyes of A cannot see something different than the camera on his head, I ask again: what will E see on Earth through this camera? 

They see themselves move at regular speed. Ice said that over and over and over again. And I've told you that your opinion otherwise is wrong. At some point you have to ask yourself if you've found a paradox within Minkowksi geometry. A branch of pure mathematics. Or whether (more likely) you haven't learned Relativity correctly. And then you can ask yourself if you want to learn. And if so, then you can read my answer.
Literally you could take images at sixty frames a second at E and send them out to A a light year away. Then A gets them and puts them on screen, records the image and their own clock and them watching both and immediately sends video at sixty frames a second (A second) back to E. E gets the images back two years after sending them and notices that they sent 60 frames every second but only get one back 12 times a second. And each of those frames will show the clock on A advancing one 1/60 of a second hand but will show five frames from E having arrived between frames.

Since E cannot see themselves moving "fast" or "normal" at the "same" time implies there is a paradox.

E doesn't see themselves fast. They see themselves faster than A.
A: @Timaeus I am sorry to say but you contradict yourself. First you agree by saying that "Yes. That's what happens" when are referring to: "A will see on his monitor E moving as in a FAST forwarding movie" and then few sentences later you say "No one (other than maybe you) ever claimed that they (E) would see themselves moving fast". I don't think there is a more clearer example of a contradiction.
Then you continue: "When I say that A sees the movie from E playing like a fast movie, I mean that they see the clocks in the movie tick at five times the rate they see their own clocks tick. I don't mean anything magical like that they can tell if movies are fast, only that they can tell they are faster than their own clocks."
First your description is incorrect and again seems to lack the understanding of whom is A and E. A is short for Astronaut so if you refer to him you address it at singular not by "they" which brings total confusion to your statement. In physics and mathematics the language should be used very precise without any ambiguities.
But lets say you refer to E as "they" so E seeing their clocks on the monitor inside the ship tick at 5 times the rate they their own clocks tick is exactly as saying that E see themselves taking 5 steps more on the monitor than they do back on Earth albeit they walk 5 times faster exactly as in a fast forwarding movie as you (and I) agreed. And here AGAIN lies the paradox since E seeing themselves walking 5 times faster on the monitor inside the ship through the camera on A's head while on Earth they walk at "normal" speed would be like seeing themselves in a mirror walking 5 times faster than they actually are which would be impossible.
In the end of course that I don't claim that I "discovered" the paradox that like a perpetuum mobile in Thermodynamics would put into question relativity. I am just saying that there is an APPARENT paradox that needs explaining, and unfortunately your arguments don't seem to do it.
PS: Another insight that I had, and possible explanation would be that since everything inside the ship is affected by the time dilation, including the monitor, E would still see themselves walking at normal speed on the screen because the slow down in the monitor intrinsic functioning circuitry would compensate for the difference in pace that A and E should see from their respective reference frames. 
Since E would see A almost "frozen" or as in a slow motion replay if looking at him through a camera inside the ship, similarly the images on the monitor screen would appear to "move" more slowly for E therefore compensating for the fast forward movie like motion that A would see on the screen in his reference frame. If this explanation is correct, then in the end E would see themselves at normal speed on the monitor screen while A would see E as in a fast forwarding movie as it should be, solving the apparent paradox.
