Learn Farsi: با اینکه کتاب را خواندیم، چیزی نفهمیدیم. - Even though we read the book, we did not understand anything.

Unlike words for "wanting" or "must" which trigger the subjunctive mood, bâ inke (even though) deals with real facts. Since it is a fact that we actually read the book, we just use the regular indicative past tense.

I know chiz means "thing". How does chizi become "anything" in this sentence?

The -i at the end of chizi makes it indefinite, meaning "a thing". When you pair "a thing" with a negative verb like nafahmidim (we did not understand), it translates smoothly to "anything" in English, as in "we didn't understand a thing."

What is the exact role of râ after ketâb (book) here?

râ is the definite direct object marker. It shows that the action was done to a specific, known book. That is why the sentence translates to "the book" rather than just reading "a book" or reading in general.

In Farsi, can I add a word for "but" before the second half of the sentence?

No, just like in English, when you start a sentence with bâ inke (even though), you do not use a word for "but" (like vali or ammâ) in the second half. The comma does all the connecting work for you.