Is one permitted to say Tehillim in פסוקי דזמרה, or while waiting for the ש”צ before/after ישתבח?
No, as it isn’t part of the set template of davening. See also here and in the sources quoted: https://asktherav.com/17257-can-i-learn-verbally-between-yishtabach-and-kaddish/ Sources: באר היטב נא, ג. ומפורש באבודרהם שאין להוסיף מזמורים…