Following the hint of Geordie Williamson, I contacted Daniel Tubbenhauser, one of the authors of the paper on categorification. With his agreement, I put here his answer:

"These are actually very interesting questions, in particular from the viewpoint of categorification.

Let me try to summarize what I know about your questions. First things first, we need to distinguish between "positive" and "positive integral" - the later is the one being reachable from the viewpoint of categorification. I can't say much about the "positive" case, but we will get there. (Basically, pi is positive, but I do not think it comes up "very natural" from a categorification point of view.)

Ok, let us start with your questions.

The second question is easy to answer: I certainly have never seen any reference discussing/calculating how the simple characters of KL basis elements look like - left aside positivity or positive integral properties. Obviously this doesn't imply that there are no references, but I can only help you with the calculations I did myself, some of which I explain below. However, there is one family of exceptions. As Geordie mentions, for anything related to cell representations there is a vast literature on the subject and positive integral properties, most notably via categorification. The recent adorable book on Soergel bimdoles in Geordie's answer covers a good part of this flavor of the theory.

Let me now try to answer your other two questions.

In any computation I ever made the simple characters of the KL basis acted positively. But this comes with a huge catch: I stopped doing these calculations very quickly since in everything except type A you will immediately hit a wall, namely non-integral coefficients turning up in the simple representations.

Here is an explicit example for the dihedral group of order 2n. This group (and its Hecke algebra, of course) has its natural, a.k.a. rotation, representation which is of dimension 2. In matrices, the two simple reflections $s,t$ act as

$$s \mapsto \begin{pmatrix} \cos(2\pi /n) & \sin(2\pi /n) \\ \sin(2\pi /n) & -\cos(2\pi /n) \end{pmatrix}, \; t \mapsto \begin{pmatrix} \cos(4\pi /n) & \sin(4\pi /n) \\ \sin(4\pi /n) & -\cos(4\pi /n) \end{pmatrix} .$$

There is basically "no way" that compositions of these have integral traces. And indeed, if you would feed these matrices into the corresponding sums defining the KL basis elements in these types (summarized in Elias' dihedral paper https://arxiv.org/abs/1308.6611), then you will see that these have non-integral characters. They have non-negative characters, you get something like $\frac{1}{2}(1+\sqrt{5})$ for $n=5$, for example. I do not know whether that is a "law of small numbers coincidence" whether this holds in general.

For the same reasons as above (non-integral coefficients), I do not have any answer to the question of how to categorify simple representations. With the current technology (which is mostly about natural or integral decategorifications) this seems to be out of reach. Sadly, my paper Geordie mentioned also doesn't help because decategorifications of simple 2-representations are not simple, again because I do not know how to categorify expressions such as $\frac{1}{2}(1+\sqrt{5})$. The decategorifications of the simple 2-representations are indecomposable representations over the integers, which, depending on your field, might decompose further into non-integral representation. The above is an example of such, it turns up if you decompose the cell representations for the middle cell in dihedral type. "

Following the hint of Geordie Williamson, I contacted Daniel Tubbenhauer, one of the authors of the paper on categorification. With his agreement, I put here his answer:

"These are actually very interesting questions, in particular from the viewpoint of categorification.

Let me try to summarize what I know about your questions. First things first, we need to distinguish between "positive" and "positive integral" - the later is the one being reachable from the viewpoint of categorification. I can't say much about the "positive" case, but we will get there. (Basically, pi is positive, but I do not think it comes up "very natural" from a categorification point of view.)

Ok, let us start with your questions.

The second question is easy to answer: I certainly have never seen any reference discussing/calculating how the simple characters of KL basis elements look like - left aside positivity or positive integral properties. Obviously this doesn't imply that there are no references, but I can only help you with the calculations I did myself, some of which I explain below. However, there is one family of exceptions. As Geordie mentions, for anything related to cell representations there is a vast literature on the subject and positive integral properties, most notably via categorification. The recent adorable book on Soergel bimdoles in Geordie's answer covers a good part of this flavor of the theory.

Let me now try to answer your other two questions.

In any computation I ever made the simple characters of the KL basis acted positively. But this comes with a huge catch: I stopped doing these calculations very quickly since in everything except type A you will immediately hit a wall, namely non-integral coefficients turning up in the simple representations.

Here is an explicit example for the dihedral group of order 2n. This group (and its Hecke algebra, of course) has its natural, a.k.a. rotation, representation which is of dimension 2. In matrices, the two simple reflections $s,t$ act as

$$s \mapsto \begin{pmatrix} \cos(2\pi /n) & \sin(2\pi /n) \\ \sin(2\pi /n) & -\cos(2\pi /n) \end{pmatrix}, \; t \mapsto \begin{pmatrix} \cos(4\pi /n) & \sin(4\pi /n) \\ \sin(4\pi /n) & -\cos(4\pi /n) \end{pmatrix} .$$

There is basically "no way" that compositions of these have integral traces. And indeed, if you would feed these matrices into the corresponding sums defining the KL basis elements in these types (summarized in Elias' dihedral paper https://arxiv.org/abs/1308.6611), then you will see that these have non-integral characters. They have non-negative characters, you get something like $\frac{1}{2}(1+\sqrt{5})$ for $n=5$, for example. I do not know whether that is a "law of small numbers coincidence" whether this holds in general.

For the same reasons as above (non-integral coefficients), I do not have any answer to the question of how to categorify simple representations. With the current technology (which is mostly about natural or integral decategorifications) this seems to be out of reach. Sadly, my paper Geordie mentioned also doesn't help because decategorifications of simple 2-representations are not simple, again because I do not know how to categorify expressions such as $\frac{1}{2}(1+\sqrt{5})$. The decategorifications of the simple 2-representations are indecomposable representations over the integers, which, depending on your field, might decompose further into non-integral representation. The above is an example of such, it turns up if you decompose the cell representations for the middle cell in dihedral type. "