The docs for lightsail suggest that it can be connected to most AWS services.
Is it possible to have a load balancer distribute to different lightsail instances and to automatically launch new lightsail instances if I'm getting high traffic? Or is this purely designed for people testing things out, so if we want scaling we need to go the full EC2 route?
If this is possible can we also use Amazon SSL with the LB to force all traffic to these instances via https?
As of 29th November 2017, you can now use load balancer with Lightsail.
See https://aws.amazon.com/about-aws/whats-new/2017/11/amazon-lightsail-adds-load-balancers-with-integrated-certificate-management/
For this use case, you'll need to use the "real" EC2. Lightsail is designed for one-off servers.
What you might want is Docker deployed to Amazon ECS on Fargate. This will autoscale as needed and you won't have to manage physical or virtual server resources.