Learn Turkish: Bir kere daha söyler misin? - Could you say it one more time?

You could, but it changes the meaning! Bir kere means "one time". If you use the distributive form birer kere, it means "one time each". You would only use birer kere daha söyler misiniz? if you were talking to multiple people and asking them to each repeat what they said.

Why is the aorist tense used here (söyler) instead of the present continuous (söylüyor)?

In Turkish, the aorist tense (indicated by the -r in söyler) is the standard way to make polite requests, translating well to "would you" or "could you" in English. If you asked Bir kere daha söylüyor musun?, it would literally mean "Are you saying it one more time right now?", which sounds factual rather than polite.

Why does daha come after bir kere? In English we say "one MORE time".

When daha is used to mean "an additional amount" or "extra", it comes after the noun phrase it modifies. So, "one more time" is built as "one time more" (bir kere daha). You will see this pattern everywhere, such as bir çay daha (one more tea) or iki kişi daha (two more people).

Could I use the verb demek (to say) instead of söylemek here?