If a Robot system with an advanced A.I. operating system had the 'ability' to 'do' 'undirected' or 'semi-undirected' introspection and 'form' all sorts of 'thought' structures about any subject matter (that didn't directly have something to do with important 'operating' functions) could these not-necessarily important 'thought structures' and the 'managing' of such cause something analogous to psychological problems in the robot?
Depending on the definition of "psychology" and "psychological problems" sure. The APA defines psychology as "the scientific study of the behavior of individuals and their mental processes." If we take this to only apply to biological systems (as I am sure many researchers do) then no, because the robot is not a biological system and therefore we can not study it under the heading of psychology and we would probably study it under something more akin to "engineering". However, if one is happy applying the same to robots, as I assuming you are, then yes.
Assuming the latter, and if you think that a being's "'ability' to 'do' 'undirected' or 'semi-undirected' introspection and 'form' all sorts of 'thought' structures about any subject matter" is a cause of psychological problems, and we are further assuming that a robot could do this, then yes, the robot will have psychological problems by definition. (Is there direct evidence suggesting these are causes though?)