The basic function of the -te form is to connect two clauses in the temporal order. However, it can have many functions, for instance, cause/reason of the main clause, the means of the main clause or adversative conjunction. The meaning differs depending on the relation between the two clauses. The key is that the relation must be clear to use the -te form in any cases. The reason why 忘れて can't express the cause/reason of the following clause 明日持ってきてもいいですか is because the relation is unclear. (Incidentally, 宿題を忘れて、明日持ってきてもいいですか sounds like as if you are asking if it's OK tomorrow you will forget your homework and bring it, i.e., the construction of the sentence is [宿題を忘れて、明日持ってきて]もいいですか.)
As for the function of the -te form 忘れて, read more carefully the textbook. It explains In the second example (= 教科書を忘れて、すみません。), the te-form describes the situation for which the apology is made.
Again, you need to use another expression you learned in the lesson to explain the reason.