Learn Polish: Mój szef dał mi radę, ale nie dał mi nagrody. - My boss gave me advice, but he did not give me a reward.

We use the Dative case (like mi) for the direct recipient of an action, especially with verbs of giving, showing, or telling. You use dla plus the Genitive case when you are doing a favor or getting something intended for someone (like buying a gift for them). Since the boss is directly giving the advice and reward to you, the Dative mi is the correct choice.

Why is the first given object radę in the Accusative case, but nagrody is in the Genitive case?

This is a great example of the Genitive of negation. In the first half, the verb is positive (dał), so its direct object (rada) takes the standard Accusative case (radę). In the second half, the verb is negated (nie dał). In Polish, whenever an Accusative direct object is negated, it automatically switches to the Genitive case (nagrody).

Could I use the long-form pronoun mnie instead of mi?

Not naturally in this context. mi is the short, unstressed form of the Dative pronoun. You should use it in most situations unless you are placing the pronoun at the very beginning of a sentence, or you want to strongly emphasize it (like saying he gave it to ME, not to someone else). Here, the short form mi is perfect.

Is there a strict rule for where the short pronoun mi goes in the sentence?