What does it actually mean to say that time "speeds up" or "slows down"? I was thinking lately about relativistic time dilation and time passing differently on other planets (which both, eventually turned out to be caused by the same principle).
Now, quite naturally (as it seems to me, at least =) I came to the question stated in the question title. Here is what I mean: time is measured in different units, which all stem from a second (which is a SI base unit). According to wikipedia, as one (probably) would expect (italic is mine):

... the historical definition of the unit (second) was based on this division of the Earth's rotation cycle ...

It is quite logical, since initially most of the things we measured time for were directly related to the Earth and it's rotation (i.e. ships leaving at the exact time etc.).
Following the quote above:

... the formal definition in the International System of Units (SI) is a much steadier timekeeper: 1 second is defined to be exactly "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom" (at a temperature of 0 K). Because the Earth's rotation varies and is also slowing ever so slightly, a leap second is periodically added to clock time to keep clocks in sync with Earth's rotation.

With that said, moving to my questions now:


*

*What does it actually mean to say that on planet A time is twice as fast as on planet B? Does it mean that while 9,192,631,770 "periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom (at a temperature of 0 K)" will occur on planet B — on planet A 18,385,263,540 such radiations will occur?

*Are the radiations in Caesium-133 are consistently relative to the properties of all other elements? Basically what I am asking here is: if we were to change the said radiation of caesium-133 in the whole universe, then the physical world, as we know it, would completely change (if not cease to exist), is this correct?

*Any stable (under given conditions) property of any element could have been selected as a basis for second, and Caesium-133 was picked up, pretty much, arbitrary, am I right in stating this?

 A: 
What does it actually mean to say that on planet A time is twice as fast as on planet B?

It is a statement about observers, for the observer in B time is twice as fast on planet A. Planet A does not know it, i.e. the definition of a second is the standard one on both planets.

Are the radiations in Caesium-133 are consistently relative to the properties of all other elements?

In the frame of all involved nothing changes. It is observations on the others frames that show a different time .

Any stable (under given conditions) property of any element could have been selected as a basis for second, 

Yes , any stable radiative transition uniquely identifiable .
You might get a better intuition if you read up on the  GPS system which needs special and general relativity too to keep correct distances on earth.
A: 
Does it mean that while 9,192,631,770 "periods of the radiation
  corresponding to the transition between the two hyperfine levels of
  the ground state of the caesium-133 atom (at a temperature of 0 K)"
  will occur on planet B - on planet A 18,385,263,540 such
  radiations will occur?

The crucial word is "while". It's absolutely essential to give it an
observational meaning. To do this many more words are needed.
We must assume that on both planets advanced physics labs have been
built, and that previous communication established that the same
physics holds on them. E.g. they will have identified chemical
elements, among them caesium. They will have built atomic clocks,
following quite similar protocols. (Just as we have here on Earth, in
several labs all world around - and here we have quite a similar
problem: how can we know all those clocks run at the same rate?
Actually they don't.)
Then a cross checking campaign is started. From lab on A e.m. signals
are sent to B at intervals of 1 seconds of A's clocks. On B arrival
times (according B's clocks) are recorded. For a best comparison, the
reverse procedure is also actuated, from B towards A. 
Data analysis, jointly executed on all data, can show that in
direction A$\,\to\,$B a factor $k_{AB}$ is observed, i.e. signals sent at
1 s intervals from A are received in B at $k_{AB}\,$s. In reverse
direction a factor $k_{BA}$ is measured. 
This unmistakebly shows a difference between both planets. Should we
say that "time runs faster (or slower, according as $k_{AB}>k_{BA}$
or vice versa) on A than on B"? I wouldn't, as a matter of personal
taste. But what really matters is the observational result.
A question for you. Why both signals are required? Wouldn't A$\,\to\,$B
have been sufficient?

Are the radiations in Caesium-133 are consistently relative to the
  properties of all other elements?

I already answered in the affirmative. I said physics is the same on
both planets.

Basically what I am asking here is: [...]

Here I have no answer, mainly because I can't understand the question.

Caesium-133 was picked up, pretty much, arbitrary, am I right in
  stating this?

Not quite. I'm no metrology expert, but I believe the choice was made
in function of some favourable properties of various kinds. Above all,
I would say, on better quality of atomic clocks built with that
element.
