In German, when is it appropriate to use "komme", and when is it appropriate to use "stamme"?

No, they are not simply interchangeable. The word "stammen" comes from "Stamm" (trunk). You use "stammen" to tell where you are originally from, where you was born and grew up (not necessarily all your childhood, but at least the first years).

"Ich stamme aus Deutschland." means you passed the first years of your life there.

"Ich komme aus Deutschland." could mean the same, but is more general. It can also mean you passed a few years there or, in another context "Ich komme gerade aus Deutschland." means you passed your holiday there.

You see, "stammen" is more profound. My cousin was born in Italy. When she was 2, her parents moved to Germany. "Sie stammt aus Italien." and "Sie kommt aus Deutschland." ;-)

stammen = to hail from a place

kommen = come

So they are interchangeable when you are talking about were you're from originally.

"Ich komme aus Deutschland" = "Ich stamme aus Deutschland."

When you are on vacations in Germany and return home you can only say

"Ich komme aus Deutschland" because you are coming from there in that moment

"Stammen" is more formal than kommen.

For example if you say: "Ich stamme aus Deutschland."

Kommen has a double meaning, it not only means that you come from a specific place but also directly translated come for dinner or come over for a visit.

In that case you can't say "stammen"

There is a word which might give you a clue. It is "Stammbaum" (family tree). Everything that has to do with ancestors, is "stammen".

"kommen" has about the same meaning as "to come" in English I think.