Learn Finnish: He ovat kotona, mutta me olemme ulkona. - They are at home, but we are outside.

Yes! When mutta connects two complete thoughts that both have their own subject and verb (like he ovat and me olemme), you always put a comma before it in Finnish.

Does the word mutta change the word order in the second half of the sentence?

No, it doesn't. Finnish coordinating conjunctions like mutta (but) and ja (and) are very straightforward. They do not change the word order of the sentence that follows them. You just use normal subject-verb order: me olemme (we are).

Why do both kotona (at home) and ulkona (outside) end in -na?

The -na ending is used here to show a fixed state of location. While Finnish usually uses other endings (like -ssa or -lla) to say "in" or "at" a place, words like koti (home) use this older -na ending to mean "at home." Kotona and ulkona are very common everyday expressions that are best memorized as a matching pair.

Since the verb olemme already tells us the subject is "we", can I drop the word me?

You could grammatically drop me (we), but you cannot drop he (they) because third-person pronouns are always required. However, keeping both pronouns is the best choice for this specific sentence. Because you are comparing two different groups (they are inside, but we are outside), keeping the pronouns strongly emphasizes this contrast.