You are asking many questions here, about which much is written down in the rather large literature of the subject. Maybe I can point to some of the answers.

In the 1955 paper of J.A. Green on finite general linear groups, the results were heavily combinatorial and left open quite a few questions about special linear and other groups of Lie type. Green's own students Srinivasan and Lehrer made significant progress on Sp$_4$ and on special linear groups, over finite fields. But here the results get far more complicated, with for example many fractions occurring in degree polynomials. This is true also for the finite groups of type G$_2$, treated by Chang and Ree, even though the ambient algebraic group in this case has a (trivial) connected center. This connected center case was first worked out most thoroughly later on, but by now the efforts of Lusztig (and others) have made it possible in principle to compute all degrees of irreducible representations for finite groups of Lie type.

An exposition by T.A. Springer of some of this including Green's work may be most useful to you. This arose from the special IAS year 1968-69, published here as Part D. Note in particular the last part of I, in which degrees of irreducible representations of finite general linear groups are expressed explicitly as polynomials in $q$. Ian Macdonald also participated in that special year (and later wrote up a version of Green's work in his book on symmetric functions); he formulated some conjectures about what should occur for other groups of Lie type, which were then essentially proved by Deligne and Lusztig in their fundamental 1976 paper. The key point in all types is the determination of the elusive "cuspidal" or "discrete series" characters. (A helpful textbook source is the 1985 treatise by Roger Carter here; but afterwards Lusztig made many further advances.)

You are asking many questions here, about which much is written down in the rather large literature of the subject. Maybe I can point to some of the answers.

In the 1955 paper of J.A. Green on finite general linear groups, the results were heavily combinatorial and left open quite a few questions about special linear and other groups of Lie type. Green's own students Srinivasan and Lehrer made significant progress on Sp$_4$ and on special linear groups, over finite fields. But here the results get far more complicated, with for example many fractions occurring in degree polynomials. This is true also for the finite groups of type G$_2$, treated by Chang and Ree, even though the ambient algebraic group in this case has a (trivial) connected center. This connected center case was first worked out most thoroughly later on, but by now the efforts of Lusztig (and others) have made it possible in principle to compute all degrees of irreducible representations for finite groups of Lie type.

An exposition by T.A. Springer of some of this including Green's work may be most useful to you. This arose from the special IAS year 1968-69, published here as Part D. Note in particular the last part of I, in which degrees of irreducible representations of finite general linear groups are expressed explicitly as polynomials in $q$. Ian Macdonald also participated in that special year (and later wrote up a version of Green's work in his book on symmetric functions); he formulated some conjectures about what should occur for other groups of Lie type, which were then essentially proved by Deligne and Lusztig in their fundamental 1976 paper. The key point in all types is the determination of the elusive "cuspidal" or "discrete series" characters. (A helpful textbook source is the 1985 treatise by Roger Carter here; but afterwards Lusztig made many further advances.)

[ADDED] It's not clear how far one can conceptualize the polynomial result, but at least for the induced representations which give irreducibles it's obvious how this property follows recursively. The cuspidal representations seem more mysterious, along with many others which are unipotent (among those which arise from decomposing induced representations non-trivially), I don't know how to account conceptually for the degrees to be given by polynomials in $q$. Of course, these degrees do have to divide the group orders, which themselves are polynomials in $q$.