Let $p$ be an odd prime number. Can a finite simple group have a conjugacy class with $2p$ elements?
I believe the answer is no. A group with a conjugacy class of degree $2p$ has a transitive permutation action of degree $2p$ on that class, and the stabilizer is the centralizer of an element.
If this action is imprimitive, then the blocks have size $p$ or 2. Wtih blocks of size $p$ there would be a subgroup of index 2, contradicting simplicity. With blocks of size $p$, the group would act faithfully on the blocks. Burnside proved that a permutation group of prime degree $p$ is solvable or 2-transitive. The finite 2-transitive permutation groups are now known (using the classification of finite simple groups). They are listed, for example, in
PETER J. CAMERON, FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS, BULL. LONDON MATH. SOC , 13 (1981),1-22
and it can be checked that none of the examples of prime degree have point stabilisers that have a subgroup of index 2 with nontrivial centre.
So suppose that the action on the conjugacy class is primitive. In the Cameron paper cited above, a result of Wielandt is mentioned that says that primitive permutation groups of degree $2p$ either have rank 3 or are 2-transitive. Using the classification of finite simple groups, it has been checked that the only rank 3 examples are $A_5$ and $S_5$ in degree 10, and $A_5$ does not have a class of degree $2p$. For 2-transitive groups, we can again check the list and observe that none of the groups has point stabilizer with nontrivial centre.