I know that the only simply primitive permutation groups of degree $2p$, where $p$ is an odd prime, are $A_5$ and $S_5$. I want to know that: Is there a complete list of simply primitive permutation groups of degree $2p^2$, where $p$ is an odd prime?
For the sake of getting this off the unanswered stack... Yes, all such actions are known.
By studying O'Nan--Scott--Aschbacher, one sees immediately that a group $S$ that has a primitive action of degree $2p^2$ must be almost simple. Now one refers to this paper:
Cai Heng Li and Xianhua Li, MR 3210408 On permutation groups of degree a product of two prime-powers, Comm. Algebra 42 (2014), no. 11, 4722--4743.
The main result of this paper includes as a special case the primitive actions of degree $2p^2$. One can now compare this to the lists of $2$-transitive actions to conclude which are simply primitive.
Recall: a simply primitive action is a primitive action that is not 2-transitive. Note that the paper above is impressively general: one can use it to get a classification of simply primitive actions of degree $p^a q^b$ (with $a,b$ positive) -- note that in this general setting one also has product action type groups, not just almost simple groups.