Intro (tldr-ish):

In the meantime, in the literature I dug up the formulae not only for the dimension D of a $G_2$ module, but also its quadratic Casimir C2 (eigenvalue). After some playing, I obtained a remarkable symmetry:

$D(G_2(i,j))=([3+3*i]*[1+j]*[4+3*i+j]*[5+3*i+2*j]*[6+3*i+3*j]*[9+6*i+3*j]) /([3]*[4]*[5]*[6]*[9])$

$W(G_2(i,j))=(]3+3*i[*]1+j[*]4+3*i+j[*]5+3*i+2*j[*]6+3*i+3*j[*]9+6*i+3*j[) /(]3[*]4[*]5[*]6[*]9[)$

where [x] denotes quantum integer $x_q$ and ]x[ $r^{x^2}$. W is the writhe factor, which is $q^{C2}$ (My definition - C2 is obviously defined only up to a constant factor, virtually no two people agree on the factor, and you have to retrofit $r=q^l$ so that the formula comes out right).
Example:

$D(G_2(1,0))=1+q^{10}+q^8+q^2+q^{-2}+q^{-8}+q^{-10}$

$W(G_2(1,0))=r^{144}=q^6$ so $l=24$. (But all Casimirs are OK after that.)

I computed the analogous formulae for $A_n$ and again, they have identical form. Then I went berzerk and computed the Casimir for all simply laced Lie algebras. Behold:

Laced[Z_,L1_,L2_,L3_]:=Sum[(1+i+j+k+Z+Sum[L1[[ii]],{ii,1,i}]+Sum[L2[[jj]],{jj,1,j}]+Sum[L3[[kk]],{kk,1,k}])^2,{i,0,Length[L1]},{j,0,Length[L2]},{k,0,Length[L3]}]-(Length[L2]*Length[L3]-1)*Sum[(1+io-iu+Sum[L1[[i]],{i,iu,io}])^2,{io,1,Length[L1]},{iu,1,io}]-(Length[L1]*Length[L3]-1)*Sum[(1+jo-ju+Sum[L2[[j]],{j,ju,jo}])^2,{jo,1,Length[L2]},{ju,1,jo}]-(Length[L1]*Length[L2]-1)*Sum[(1+ko-ku+Sum[L3[[k]],{k,ku,ko}])^2,{ko,1,Length[L3]},{ku,1,ko}];

How this MATHEMATICA snippet works: Z is the weight at the central 3-junction and L1,L2,L3 the weights of the legs (as lists) going outward. So e.g. Laced[0,{1,0,0},{0,0},{0}]=165 (dim 27664).

Phew :-)

Question 1: Of course I now like to have the dimension formulae for $E_k, k=6,7,8$ (quantum or not), but in a paper of 2011 ("Partial differential equation approach to F4") I found a formula for $F_4(0,0,i,j)$ which somewhat implies that this is so complicated it wasn't done explicitly yet (as otherwise, why didn't he give the whole $F_4(g,h,i,j)$ formula?). Jim Humphreys suggested Tits and Fulton-Harris, but I only found $B_3$ in the former and nothing in the latter.

Question 2: I have no doubt that the snippet also works for the relevant affine Lie algebras, anyone knows a source for their Casimirs? Gould's paper looks very relevant but as usual, only general formulas are given.