Why would 'mluvíš a poslouchám' not be acceptable? Isn't the meaning clear from the differing conjugations?
Here, without the pronouns, the contrast dissapears and therefore they can't be omitted.
You can say though "Mluvíš a já poslouchám." But the "já" has to be there to express the contrast that you are doing something while I am doing something else.
TBH, this answer doesn't help me understand at all. I need more explanation.
Why isn't "říkáš a ja poslouchám" accepted?
Říkat normally requires an object (the thing one is telling).
Mluviš, a poslouchám já
And was marked wrong - is the word order weird?
Quite so. If you stress "já" this way, you should also include and stress "ty".
The program gave me the wrong answer=
You used the wrong word.
povídáš a já poslouchám.
"povídáš a já poslouchám" is accepted
Can it be ty mluvíš a ja slyším
Of course both verbs are derived from "sluch", which means "(the sense of) hearing".