Syntax and Typography of Atomic Orbitals
Question 1. The Green Book p. 32 defines the letters s, p, d, f, g, h, i, k ... (omitting j). They have to be typeset upright.
  
  How far does this "alphabet" reasonably go? (Wikipedia goes up to i).



Section 2.6.3 of that version of the Green Book shows the term symbols (Wikipedia) which are capital letters S,P,D,F,G,H,I,K,... and used to indicate the electronic configuration of the valence electrons and for excited states. 
The lower case letters are used for the azimuthal quantum number (ℓ) which is one of the four quantum numbers for an electron in an atom. You can typeset $99.9~\%$ of chemistry of even more using only the letters s, p, d and f, corresponding to the left-hand main group elements, the right-hand main group elements, the transition metals and the lanthanides/actinides, respectively. Everything above f is and will be for quite some time not relevant to chemistry outside a very small circle that would know how to deal with stuff.



  Question 2. On the same Greek Book page, electron configurations are explained, e.g. $\mathrm{(1s)^2 (2s)^2 (2p)^1}$.
  
  Many other sources write it without parenthesis: $\mathrm{[Kr]\mskip2mu4d^{10}\mskip2mu5s^1\mskip2mu5p^3\mskip2mu5d^4}$
  
  Question a: What values can the left-side number take? (At Wikipedia, I see 1 to 7.) Can it be 2-digit?
  
  Question b: What values can the right-side exponent take? (At Wikipedia, I see 1 to 10.)



This has two sub-questions.


By a similar reasoning as presented in the answer to 1, I think 8 will be the limit until the not-so-close future. But there is nothing intrinsically preventing it from becoming two-digit at some distant point.
The superscript can take values of up to $4l+2$, where $l$ can be considered ‘computer science counting’ of the letter sequence s, p, d, …; so for s $l = 0$ and thus $4l+2=2$; likewise for p ($l=2$) the value is maximum $4l+2=6$ etc. Note that $4l+2$ is actually $2(2l+1)$: each subshell gives up to $2l+1$ orientation possibilities that each can be populated with two electrons.




  Question 3. I think I did not come across the configuration with one-electron orbitals (in Greek). $\mathrm{(1\sigma)^2 (2\sigma)^2 (3\sigma)^2 (1\pi)^1}$. (The Greek letters should be upright, but that's a drawback of MathJax.)
  
  Would this be useful? Or not needed?



Yes, this is useful, especially for discussing molecular orbitals. The Greek letters you want to be aware of are &#x3c3;, &#x3c0; and &#x3b4;. Maybe a fourth will be added soon corresponding to f orbitals (&#x3c5;?) but I don’t see that coming anytime soon.



  Question 4. The examples introduce another notation, "electronic states"  $\mathrm{\cdot\cdot\cdot (2a_1)^2 (1b_2)^2 (3a_1)^2}$
  
  a: What values could the letter take? b: The left-side number? c: The subscripts? d: The right-side exponent?



I’m going to answer that as one.  

This notation uses irreducible representations to define orbitals. The bit between the initial number and the superscript can be any of the symbols that appear in Little Bobby Tables (courtesy of Orthocresol). They each represent a type of orbital with a certain symmetry.  

Orbitals of the same irreducible representation (i.e. the same centre) are numbered from lowest to highest energy — so in principle the sky is the limit. Practically, however, you will be hard-pressed finding three-digit examples.  

The exponent can at maximum be $2n$, where $n$ is the number in the first column of the respective character table (the E column). This number corresponds to a certain symmetry’s degeneracy (i.e. so many orbitals of the same energy) and each orbital can be populated by two electrons.



  Question 5. From Wikipedia, I learned that there are notations IUPAC does not mention: $\mathrm{p_x}$, $\mathrm{p_y}$ and $\mathrm{p_z}$. (All subscripts should be upright, because they are labels, no variables.)
  
  But it does not end there. Here and here I saw: 
  $\mathrm{d_{xy}, d_{xz}, d_{yx}, d_{x^2{-}y^2}, d_{z^2}}$
  This was confirmed by this book. There are alternative styles 
  here ($\mathrm{{p_x}^1}$).
  and
  here ($\mathrm{p_{x1}}$).
  
  a: I guess the $\mathrm{d_{z^2}}$ notation (with index z-square) is correct?
  
  Oh, I just found this book that also includes $\mathrm{4f_{x(x^2{-}3y^2)}}$ and $\mathrm{4f_{y(3x^2{-}y^2)}}$.
  
  b: Are there other indexes possible than x, y, z?
  
  c: Can it be just squares? Or are other index-superscripts possible?
  
  d: What about the 3 in the last examples? Could this also be any other numer?
  
  e: Could there be any other forms of indexes?
  
  f: Typographically, what does this dash stand for? Is it a hyphen or a "transition between a higher energy state and a lower energy state"?



(Somebody please add and correct this.)

It was my assumption that the axis directions should be italicised but I don’t know. To discuss this, please go to the break-out discussion for this question.

p orbitals will always have $x$, $y$ or $z$. d orbitals will have one of the five you noted ($\mathrm{d}_{z^2}$ is correct — as I said, I am not sure on the upright versus italic question but I would have assumed italics). The f orbitals themselves can be labelled by various different polynomials in $x$, $y$, $z$, and $r$. See these two web pages: 1, 2. In general the subscript is just a polynomial.

Which symbols and exponents can appear here is fixed already by the results of the hydrogen-like orbitals — again, for most of chemistry d orbitals are sufficient and you can cover $99.9~\%$ by including f orbitals.

The ‘dash’ is a mathematical minus-sign operator. Yes, it truly is $x^2$ minus $y^2$.



  Question 6. Finally, there is orbital hybridization, which will lead to notations like $\mathrm{sp, 2sp, sp^2, 2sp^3, s^{0.5}p^3, s{-}s, sp^3{-}sp^3}$.
  
  a: Can the superscript be any other value than 2 and 3? Can the "base" be any other thing than s, p and sp?
  
  b: Is $\mathrm{2sp^3}$, is it $\mathrm{2s}$ and $\mathrm{p^3}$ or is sp one ... thing with a 2 before and a 3 in superscript.
  
  c: Is it a hyphen or a "transition between a higher energy state and a lower energy state"?



Sub questions.


Yes. You have $0.5$ as a superscript in your own examples. But, as posts by Martin and Ron somewhere point out, in $\mathrm{sp}^x$ the superscript $x$ can, in principle, be any number.
$\mathrm{2\,sp^3}$ is $\mathrm{\{2\}\{sp^3\}}$ — and there could potentially be another superscript making it $\mathrm{(2sp^3)^2}$.
I don’t know what that hyphen is supposed to be.