Learn Croatian: Doktore, možete li mi pomoći?

It is in the vocative case, which is the form you use when directly calling or addressing someone. For most masculine nouns ending in a regular consonant like -r, you add the ending -e to form the vocative.

Since we are using the formal you (možete), why doesn't the vocative ending for doktor change to show respect?

The vocative ending is strictly based on the noun's gender and its final letter, not on politeness. Even though you are addressing the doctor formally with the verb možete, the noun doktor will always take the standard -e ending when you call out to him.

Why are the little words li and mi placed where they are?

li (the question particle) and mi (to me) are clitics, which are short words that must go in the second position of a sentence. A word used to get attention, like Doktore, is separated by a comma and doesn't count as the first position. The core sentence starts with možete, so the clitics tuck in right after it.

Why do we use mi (to me) instead of the direct object form me?

The verb pomoći (to help) always takes the dative case in Croatian. Mi is the short dative form of I. It helps to think of it literally as asking can you give help to me?.