Both of these would not express progressive actions. When you connect verbs with て-form, it would mean you finished the first action, then did the second action. 座っていて would be ungrammatical because you imply a progressive action is already done. Instead, for progressive actions, you would add ながら to the end of the ます-stem of a verb.
He was messing with his phone while walking.
For your example, however, this gets a bit more complicated. Sitting in Japanese is an instantaneous action, not a state. While you can say you were "sitting while talking" in English, 座る means the action of sitting down and is instantaneous. You cannot "read a book while performing the action of sitting down". Instead, you would "sit down, and then read a book".
So, instead of:
彼は座りながら本を読んでいた。X This is incorrect
you would say:
He sat down and read a book.