Learn Farsi: برادر بزرگ با آن عروسک قدیمی بازی می‌کرد. - The big brother used to play with that old doll.

It uses the continuous prefix می‌ (mi-). You attach it to the past stem of the verb, which is کرد (kard) from the infinitive کردن (kardan, "to do"). Because the subject is "the big brother" (3rd person singular), the past personal ending is empty or "blank". So, می‌کرد literally means "he was doing".

Why is the verb split up, and why doesn't the می‌ (mi-) prefix go at the very beginning of the word?

"To play" is a compound verb in Farsi: بازی کردن (bāzi kardan), which literally means "to do play". The first part (بازی) is just a noun. Tense prefixes like می‌ (mi-) only attach to the actual verb part (کردن), so the noun stays separate at the front.

The translation says "used to play". Could this sentence also mean "was playing"?

Yes, absolutely! The Farsi past continuous tense covers both habitual actions in the past ("he used to play every day") and ongoing past actions ("he was playing when I walked in"). Without more context, both translations are completely correct.

I don't see a connecting word between "brother" and "big", or "doll" and "old". How are they linked?

They are linked by the ezāfe, a short "e" sound that connects nouns to adjectives. You pronounce it as barādar-e bozorg (big brother) and arusak-e qadimi (old doll). The ezāfe is usually invisible in Persian writing, so you have to remember to pronounce it when you see a noun followed by an adjective!