I don't know if they do psychological tests like that I never took one :)
They can do one if you give them reason to, such as a mental condition.
In general, no, a PTSD diagnosis is a bar to enlistment, however there are rare exceptions. In the case of a linguist, for example, who speaks one or more rare languages.
It is a PDQ, because they are not likely to give a waiver for this serious of a condition