I want to provide a commercial license for somebody who is willing to pay on top of open source one (LGPL allows it, right?)
None of the FLOSS licenses are exclusive licenses. This means that as long as you're the sole owner of the project's copyright, you're free to dual-license it under some FLOSS license (including LGPL), and a commercial license.
I want to allow the usage in commercial projects (not sure what to do here, because LGPL has dynamic linking exception, but iOS is one of the platforms I'm targeting and on iOS static linking is still most popular option)
LGPL has a very broad linking exception (it covers all types of linking) that allows linking of code under LGPL to a commercial project without buying a commercial license. Your dual-licensing business model will not work with LGPL. Nor will the dual-licensing work with any permissive license. Dual-licensing only works with strong copyleft, such the GNU GPL (because some commercial clients may be willing to pay for a commercial license to escape the copyleft obligations).
I want to be capable to merge any derivatives back without any permission from somebody who made modifications without preserving any of the credits of modifier. As well I want to have my copyrights on all the code, does not matter where from it comes, merged back from a fork or written in-house.
There is no way you can do this under any real free software license.
You can ask people to turn over their copyright to you by having them sign a legal document called a Contributor License Agreement (CLA). But if they don't want to sign your CLA, they don't have to.
Basically I need a license, that states that modifier waives any claims or demands to the modifications he made.
There is no license (free or not) that does this. What you're looking for is a contract of employment. People sometimes sign these, but not without a reasonable expectation of being paid.
I want to forbid any third party to re-license the code
A copyleft license, such GNU GPL, don't allow sub-licensing, so that should be OK.
and/or change copyright and credits.
They can't remove your copyright or credit (but if it is under a FLOSS license, they can use it and fork it).
However, you can't stop anybody for copyrighting their own improvements/modifications and then refuse to sign the copyright of those over to you.
And if you introduce a CLA that suggests they sign their copyright over to you, expect that somebody takes your code and fork your project as a free/libre project that people can contribute to without signing any CLA.
Is it possible (legal?) to add CLA/CAA inside the LICENSE of the project?
It is a free country, so it is possible and even legal to destroy a good license with nonsense (this is called a "crayon license"), but it won't do what you want.
How enforceable is it in case of a breach?
Not at all. Simply, by doing this, you've created a crayon license with conflicting terms. Such things are generally not enforceable through courts at all.
If you want to enforce such strange terms, you need to have the other party sign something that is separate from the software license (e.g. a CLA, a CAA, or a contract of employment) - not just "violate" some arbitrary restriction that you want to impose on others.
The main idea is to allow all the forks and everything, but make every one who modifies the software to "sign" the CLA/CAA, not just the ones who want to merge back. So, if I would merge (find merged) LGPL and CLA/CAA into one combined license, would it be legal
It would be legal in the sense that there is no law against creating a license that does not make sense.
and enforceable in a case of breach?
The problem is that I want to monetise the software.
No, the problem is that you want to monitize somebody else's software.
There is absolutely no problem with you selling your own software under a commercial license. The problem starts when you want to take the work of others and re-license it under a different license than the license they offer their work under.
According to copyright law, you cannot do this without permission from the person who created the work.
I want to make it available free for open-source community, but I don't want any claims from any contributor (this does not mean that there will be no contributor list, though). I just want to preserve the right to merge anything back without any permission, so I want a license, that explicitly states, that modifier waives any claims and understands, that even his credit might not be preserved.
The license you ask for does not make sense.
For obvious reasons, no such license exists. And even if you did create one such a "license" yourself, it would not be enforceable unless you can get people to sign a physical paper copy of it and give that signed copy to you.