There are a number of questions regarding the free will vs. qadr juxtaposition on the site already (see e.g. this one on whether humans have free will in faith, this one about free will vs qadr, again free will vs qadr, the comments and answer to this one relate free will vs qadr to "is Satan to blame for evil"), as well as some pulling Allah's attribute of Justice into the mix (this question about whether nurture overpowers free will's effect of justifying punishment).
The problem with all of these posts is that I am unable to understand what each of them means by "free will" or "qadr"; there seem to be different schools of thought on free will according to this and this and some attempts at resolving ahadith that qadr can be changed by du'a here, but those answers do not seem to develop the notions very clearly to me, and the one on qadr seems contradictory.
Any attempt at an explanation I have seen so far fails to address the question of "does Allah give creatures with free will a power to choose in themselves?". An answer of "creatures are given an independent power of chosing" seems to contradict the idea of "la hawla wa la quwatta illa billah" (with regard to quwatta there is mention of it in 18:39, I couldn't find a source for hawla), and an answer of "Allah creates their choices and their inclinations and decisions" makes the will not "free" in any sense I can discern and does not seem to be any kind of basis for punishment (this is where free will vs qadr collides with justice).
To sum up, here are my questions:
- what understandings of "free will" exist in the schools of thought, and in what sense is will considered to be "free" here?
- what understandings of "qadr" exist in the schools of thought?
- how do the schools of thought resolve the juxtaposition of these with regard to Justice?
What the main orthodox schools of thought say is my main interest, but I'd also be interested in the opinions of extinct schools, e.g. the Muta'zilites, just in case someone happens to know.
Due to the experiences I've had with people trying to explain this: I'm not interested in posts begging the question like "of course we have free will, how else could Allah be just?" and things like that; this is exactly the point of my question, I can't make sense of these concepts taken together. Analogies seem to be largely useless in this regard, e.g. this answer (which I find pretty much useless in the context of my question) here completely ignores that Allah is the creator of everything created, while the analogy assumes "randomness" originating from somewhere else; similar criticism to the consistency (!) of what the answer says applies to the answer here, and I expressed that criticism in the comments to it. I don't think more vague analogies will prove any more useful; I'm interested in precise attempts at explaining the points of friction I outlined above.