Is it possible to attach an EBS drive to a lightsail instance (aside from the one that (appears to be) auto-attached when creating the instance)?
The way I solved this was to run an ordinary EC2 instance and use NFS to mount a volume from it on my Lightsail instance. I have not tried if EFS works. Be sure to enable VPC peering in the Lightsail account settings. With VPC peering enabled, you can use security groups to allow the private IP of the Lightsail instance to connect to your EC2 instance. For optimal performance, you should launch your EC2 instance in the same availability zone that your Lightsail instance is running in.
You can mount the NFS mount using the "fsc" mount option along with cachefilesd to keep a cache on the Lightsail instance. I don't know how much this improves performance yet.
Not right now. You can get an idea about what you can and can't do from the API reference, specifically the actions list: http://docs.aws.amazon.com/lightsail/2016-11-28/api-reference/API_Operations.html