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Also try using a CDN solution like cloudflare.com it will speed things up, and reduce your monthly bandwidth bill, so more bandwidth available for audio files.
You are correct, a basic install of Ubuntu with nginx should do the trick. The one thing to note is that should you get close to your 20GB limit it would require resizing the droplet which will require up to 30 minutes of downtime so be sure that you select a plan with enough space for your needs for a while. As for bandwidth, the $5 plan includes 1TB of transfer (inbound + outbound) on the public interface. Once you reach the 1TB limit in a month additional bandwidth would be billing at a rate of $0.02/Gb.
It is possible to mount DreamObjects as a folder on your droplet.
http://www.dreamhost.com/cloud/storage/
DreamObjects is 2.5cents per GB, so you could possibly use that to store files.
I use this method as my backups folder, and it works great for that. It costs me less than $1.00 per month and I back up several droplets every night… I wrote a tutorial on how to set it up:
https://firefli.de/tutorials/s3fs-and-dreamobjects.html
You could use the same method for AWS storage, but the price is slightly better for DreamObjects.