Learn Polish: Jestem bardzo głodny, czy możesz dołożyć mi ryżu? - I am very hungry, can you give me some more rice?

This is the Partitive Genitive. Verbs that indicate adding an indefinite quantity of something (which very often start with the do- prefix) take the Genitive case to mean some more of an uncountable noun. Using the Accusative (ryż) would sound like you want someone to add a whole, specific batch of rice, rather than just an extra portion.

What does the prefix do- add to the meaning of the verb here?

The prefix do- indicates adding to or supplementing something that is already there. In the context of food, dołożyć is the standard, idiomatic verb used to ask someone for seconds or to put another helping on your plate.

Why is the pronoun mi placed before ryżu rather than at the end of the sentence?

In Polish, short pronouns like mi act as enclitics and naturally gravitate toward the middle of the clause. Putting mi at the very end of the sentence (dołożyć ryżu mi) sounds awkward unless you are placing heavy emphasis on the fact that the rice should be given to YOU specifically.

How would this sentence change if the speaker were female?

The first clause uses the masculine adjective głodny. A female speaker would change this to the feminine form, saying Jestem bardzo głodna.