strace is useless for containers, you cant attach across container boundaries, and usually you can't install it into the container, either. Which is really sad because its such an useful tool.
Not really, you can start it from the host and trace any container process you like. On Linux the processes are in one large tree and the host has IDs for all container processes. Inside the container you may have different IDs if a pid namespace is used but from the outside these processes are in the host's process tree and have regular IDs.