Many Latin neuter nouns have the same form in the nominative and accusative. Context resolves it. Here, the verb dat needs:
Since pater already fits as subject and magistro fits as indirect object, consilium is read as the direct object even though its form could also be nominative.
Yes. Latin word order is flexible because endings show grammatical roles. This order is common and slightly emphasizes the recipient before the thing given. You could also see:
The meaning stays basically the same as long as the case endings remain unchanged.
Because magister is 2nd declension masculine:
So magistrī would mean of the teacher (genitive), not to/for the teacher (dative).
Declension tells you the pattern of endings, which is how you identify case and therefore each word’s grammatical role.