Question

In the past few days I've become increasingly intrigued by the QHE, mainly thanks to very interesting questions and answers that have appeared here. Unfortunately, I am as of yet very confused by all the (seemingly disparate) stuff I learned.

First, here are some random points that I've been able to gather

I(nteger)QHE occurs due to the presence of Landau levels
IQHE is an embodiment of topological order and the states are characterized by the Chern number that tells us about topologically inequivalent Hamiltonians defined on the Brillouin zone
IQHE requires negligible electron-electron interations and so is dependent on the presence of impurities that shield from Coulomb force
F(ractional)QHE occurs because of formation of anyons. In this case Coulomb interaction can't be neglected but it turns out an effective non-interacting description emerges with particles obeying parastatistics and having fractional charge
FQHE has again something to do with topology, TQFT, Chern-Simons theory, braiding groups and lots of other stuff
FQHE has something to do with hierarchy states

So, here are the questions

Most importantly, do these points make sense? Please correct any mistakes I made and/or fill in other important observations
How do explanations 1. and 2. of IQHE come together? Landau quantization only talks about electron states while topological picture doesn't mention them at all (they should be replaced by global topological states that are stable w.r.t. perturbations)
How do explanations 4., 5. and 6. relate together
Is there any accessible introductory literature into these matters?
Do IQHE and FQHE have anything (besides last three letters) in common so that e.g. IQHE can be treated as a special case? My understanding (based on 3.) is that this is not the case but several points hint into opposite direction. That's also why I ask about both QHE in a single question

Answer 1

Here are some comments on the points:

1) I(nteger)QHE occurs due to the presence of Landau levels

Yes

2) IQHE is an embodiment of topological order and the states are characterized by the Chern number that tells us about topologically inequivalent Hamiltonians defined on the Brillouin zone

IQHE is an example of topological order, although topological order is introduced to mainly describe FQHE. The characterization of IQHE by Chern number of energy band only works for non-interacting fermion with no impurity, while IQHE exists even for interacting fermions. So IQHE is more than the Chern number of energy band. The quasiparticles excitations in IQH states are always fermions.

3) IQHE requires negligible electron-electron interactions and so is dependent on the presence of impurities that shield from Coulomb force.

IQHE does not require negligible electron-electron interactions. IQHE exist even in the clean system with Coulomb force, if you control the electron density by gates.

4) F(ractional)QHE occurs because of formation of anyons. In this case Coulomb interaction can't be neglected but it turns out an effective non-interacting description emerges with particles obeying parastatistics and having fractional charge

FQHE occures not because formation of anyons. Fermion alway carry Fermi statistics by definition, and they are never anyons. FQHE occures because of strong interacting effects. The effective non-interacting description does not really work (for example, it fails to describe the edge states and non-Abelian states).

5) FQHE has again something to do with topology, TQFT, Chern-Simons theory, braiding groups and lots of other stuff

FQH states contain a new kind of order: topological order. The low energy effective theories of FQH states are TQFTs (such as Chern-Simons theories). The quasiparticles excitations in FQH states are anyons.

6) Hierarchy states are examples of FQH states.