1. I think I have the general sense of this, but I guess I would need more maths than I have to be able to get to grips with Hilbert space; not to mention “linear operators on the Hilbert space”. Any simplification on the Wiki article would be appreciated.
Hilbert spaces are well covered in wikipedia it's not that complex most layman tend to think in normal Euclidean space but I know you have familiar with the idea of curved space and you are familiar with latitude and longitude and curvature on earth.
It's the same problem while you sit at your desk and plot a path from London to New York you think in a straight line but you are quite aware that you are actually taking a curved path. So a Hilbert space allows for the shape of space to not be flat it's a generalized description of space it's not implicitly flat and actually it covers a range of shapes space could have that are weird and whacky.
The operators are the sorts of measurements etc and so we have to know certain things work. A few are listed in the wikipedia article such as Pythagorean theorem must hold or you aren't on a Hilbert space. Think on earths surface if we plot a huge triangle so long as you account for the curvature of the earth Pythagorean theory still holds and so we assume that of space.
Linear means exactly that it's linear ... Pythagorean theory holds without having to make some sort of algebraic adjustment for curvature which would be the other option.
Non linear option is defined like so
http://www.encyclopediaofmath.org/index.php/Non-linear_operatorSo if you want it in a simplification it is basically saying you can extend our normal geometry laws out into space and they work correctly even if space isn't flat as that is the only observational facts we have so that is what we go with. Now we extend it a bit more than just geometry to motion etc but you get the idea basically it says our patch of space works like space as a whole regardless of it's shape.
See each point spells out the basis on which it is made you can attack this point on the basis "our area of space is special" because we can't not observationally exclude that.
4. I would not be able to explain that to someone else, so I guess I don’t understand it.
Four is simply saying that the only thing we see change between the quantum mechanics and classic physics is the decoherence change. We have measured and found no other changes so in the same way we create a boundary between classic states like solid/liquid/gas (boundary condition is atom or molecule movement) it is formalising that classic physics is simply a state of quantum mechanics with a defined boundary due to decoherence.
Our ancestors probably never realized that water vapour, water and ice were all the same thing so science felt it important to document the observational fact, something you probably take for granted these days.
So it is a consistent approach water vapour becomes water and then becomes ice as the molecules slow speed number 4 is simply stating classic physics is the QM world viewed with a loss of coherence. Water vapour, water and ice are all the same thing so the classic world and the QM world are the same thing with a boundary condition.
If someone was to find something else change QM is dead. Remember all of these 12 key ideas are continually under attack scientists are trying to find pilot waves and find other things between QM and the classic physics but these have withstood all tests so far.