Can речі be understood as topics or subjects like the Russian translation of Почему вы пишете про эти речи, or does it ONLY mean things?
"Річ, речі" in Ukrainian means "thing, things" or "belongings" in direct or figurative sense. Therefore, in this context it can be understood as "topics" or "subjects", too, but "речі" has a more general meaning than these two words. Your example in Russian confuses me somewhat, because "речь" in Russian means "speech" (which is "промова" in Ukrainian).
речь (ru) is also a question (as in topical), discourse, or discussion. Sorry, that's what I meant by subject or topic. So річ/речі is a "false friend" of речь/речи (ru), correct? In Russian, a "thing" is вещь or штука
I've never encountered "річ" in the meaning of "speech" (речь), so, yes, it is a false friend.
There are Ukrainian words such as речення ("sentence" as in "series of words"), речник (speaker), наречений (fiance). They may have the same origin with the Russian речь.
They certainly do--з Київської Русі! :-) Even further back than that, possibly before the Slavs became isolated between the West, East, and South