OK, here's what I've found out, in case anyone else is interested:
Yes, it's possible to stream directly from Dropbox. I have streaming set up now using Couchy Play. This works even if my computer is turned off, so it's just what I wanted. Unfortunately, my audio, even in MP3 form, take up more space than I have in my free Dropbox account (about 3.5 GB). Also, I had to give Couchy Play access to my Dropbox account. This makes me nervous, though at the moment I don't have much on Dropbox other than music.
Streaming from the computer works well using Plex. And my OneDrive has lots of storage room (1 TB, free!) But it seems that there's nothing like Couchy Play for OneDrive, yet.
Jotham, thanks for the tip about dBpoweramp. I've been using it to rip my CDs to FLAC format and it's been great. I just rip them directly into a OneDrive folder - automatic back-up!