Need to be able to download customers emails (from email-to-case). We know it is not possible to download/Save Email Messages (*.eml, *.msg, ...). This is documented here:
https://success.salesforce.com/ideaView?id=08730000000Lj02AAC

I haven't found any App on the AppExchange to be able to download/save emails.

I have submitted a Case to Premier Support. Answer: "not possible, please submit a question to our Developer Forum"