This line of auestioning is natural but needs more careful formulations to deal with the subtle things that go on for finite groups of Lie type. Presumably "maximal tori" of the finite group are meant to be the groups of rational points of maximal tori of the ambient algebraic group stable under $F$. But this already gets tricky, depending for instance on how big the field is and how close the tori are to being split over the field of definition.
It's probably easiest to answer affirmatively the broad question of whether such finite "tori" contain regular elements (in the algebraic group sense) when $q$ is large enough; but getting an optimal lower bound on $q$ might depend on the isogeny type, etc. The picture is fairly clear from the work of Cartter's student Derizotis explained briefly in Carter's section 3.8 at the end of Chapter 3. But here the algebraic group is simply connected and "simple" in the sense of algebraic groups, so the simple adjoint groups might need extra discussion. Note especially the note by Carter at the bottom of page 105, concerning the occurrence of regular elements in a given conjugacy class of the finite group. For example, in the picture of the Brauer complex for $B_2(5)$ on the next page, you can see 13 interior fixed points of $F$ which correspond to classes containing regular elements. But the corresponding tables in Srinivasan's 1968 Trans. Amer. Math. Soc. paper on characters of that finite group illustrate the fact that some of the maximal tori fail to have regular elements over such a small field even though 5 exceeds the Coxeter number here. The paper does have minor errors, but is mostly reliable.
[If I read this example correctly, it gives a negative answer to your question about nondegenerate tori of the algebraic group. But I haven't looked closely at that material.]
In any case, it's worth exploring a number of small rank groups to pinpoint what information is of most interest to you. The subject becomes quite intricate for arbitrary groups of Lie type, but the papers by Carter and Deriziotis are well worth looking at. Groups of type $G_2$ (for which an old paper by Chang and Ree computes characters and classes) are especially nice because there is only one isogeny class of groups to consider.