$\begingroup$

How can I choose the preferred website to put my questions?

I once read something along the lines "If you have to ask whether to ask on MO or math.se, likely you should ask it on math.se." I think this is a good rule of thumb.

What if someone writes his or her question down on the not preferable website?

If you post on MO and they think it is too low level, they'll close it, and maybe "migrate" it here. Since this site is for "any level" there is also not a too high level. No one will (or at least should) scold you for asking a too advanced question here. It could happen that you get no answer and/or somebody recommends that you could have asked it there. You then can still ask it there too.

Do the users of both websites have the same chance to get answers to their questions?

On the one hand, there are literally a few hundred thousand questions answered on this site that would not have been answered on MO (they could have been answered there but they would not have been answered there). On the other hand, there are likely several thousand questions answered on MO that would not have been answered here (and a reasonable fraction of could not have been answered here).

So it depends what you look at. Yet, generally, a question that would be considered as good on MO will usually be treated better and more completely than here.

Are there any tools to move the questions between the two websites? to derive less hard questions from that hard ones, and share them as parts of a question to allow the lower lever users participate also in the upper-level questions?

It is possible to move entire Q&A threads from one site to the other. This is called migration. It is not possible to move "slices" of a thread. Yet everybody is free to take inspiration from content on one site and post about it on the other site. (Of course, on should provide references as appropriate.)