Learn Farsi: ما جدا زندگی می‌کنیم.

That is right! In Farsi, many adjectives can be used exactly as they are to function as adverbs. You do not need to add an equivalent to the English '-ly'. The word جدا works for both 'separate' and 'separately' depending on its position in the sentence.

I often hear the word جداً (joddan). Is that the same word as جدا in this sentence?

They look similar and share a root, but they have different meanings and pronunciations. جدا (jodā) means 'separate' or 'separately'. The word جداً (joddan), which takes the Arabic 'tanvin' ending (the two small diagonal dashes on top), means 'seriously' or 'really'.

Why does جدا come before the compound verb زندگی می‌کنیم instead of after it?

In Farsi, adverbs of manner (words that describe how you do something) are usually placed right before the verb. Since زندگی می‌کنیم (we live) is the verb phrase, جدا sits right before it, giving us the standard Subject-Adverb-Verb word order.

How would I change this sentence to say 'We live separately from each other'?

You would use the phrase از هم (az ham), which means 'from each other'. Place it before the adverb to make: ما از هم جدا زندگی می‌کنیم (mā az ham jodā zendegi mikonim).