I can tell you that $L_2(p)$ always has an irreducible character of degree $p$ for any prime $p$. Recall that $L_2(p)$ is simply $G/Z(G)$ where $G$ is $SL_2(p)$. The character of $L_2(p)$ of degree $p$ is then simply obtained from the Steinberg character of $SL_2(p)$. This is defined, for instance, in Digne and Michel's book "Representations of Finite Groups of Lie Type". Unfortunately I don't know enough about double covers to tell you why you get the character of degree 2p. However it should be a character covering the Steinberg character.

You will sometimes see such naming conventions used, especially for finite reductive groups. For instance people sometimes write $E_6^{sc}(q)$. I would imagine the reason for the way it's written is the following. When proving certain theorems about connected reductive algebraic groups you reduce to the case where G is simple and either adjoint or simply connected. You will then often analyse each type individually. So one would start with "let G be a simple simply connected algebraic group" and then proceed to "assume that G is of type A_n" for instance.

@Demin As Jim says the simply connected groups of types $B_n$ and $D_n$ are $Spin_{2n+1}$ and $Spin_{2n}$ respectively (they are double covers of the corresponding special orthogonal groups). You might find section 1.11 in Carter's book "Finite Groups of Lie Type" a useful read as he reviews the classification of simple algebraic groups. The simply connected groups you list are simple algebraic groups and are simple as an abstract group under the conditions I gave in my answer. If this answers your question it would be good to accept the answer so people know the question is answered with

@Jim Thank you, in my haste I had forgotten (as always) that $SL_{n+1}$ is of type $A_n$. I have corrected my answer accordingly.

Just thought I'd add the following note: As an example of just how complicated this can become you might want to look at the article "Induced Cuspidal Representations and Generalised Hecke Rings" by Howlett and Lehrer. Even in this situation (which is reasonably optimal in the theory of finite reductive groups) they are left with an unknown 2-cocycle which one still has to compute to get an explicit description of the endomorphism algebra.

Hi Geordie! That I am not sure about, although it sounds plausible. Probably the person to ask about that would be Daniel. This would be useful to know if it were the case.

Thanks for your answer Will! Indeed, I had been assuming that I had indeed even an isomorphism $F^*\mathcal{F} \to \mathcal{F}$, which I should have put in the post. It is good to know that this situation is more complicated for an arbitrary local system!