I'm not an expert of LXC, but the DockerFile is really helpful to create everything you need without messing around with several configuration and files.
And think about the Docker HUB! It's amazing, don't reinvent the wheel!
You definitely can, the principle is still valid. Anyway, Route53 is extremely cheap, less than a dollar for a million queries per month, is very reliable and it integrates very well with the other AWS services, so it's definitely worth the price, IMO.
What alternative services where you thinking about?
I already have a domain registered on dreamhost and I didn't want to change nameservers. But reading about the alias concept and its cheapness, as you said, I think I'm going to migrate to R53!
Yes, we have a dedicated engineer in our team that is working on them. We are using our own technology to build and serve them. There is a bunch of them ready to be published, always on AWS for now.