Fuse ESB is using equinox as OSGi container already. However it can be configured to use another such as Apache Felix.
Yes when you deploy JBI assemblies the deployer will turn that into OSGi. In fact it all becomes OSGi, for example as well when deploying WAR files.
Yeah you can start/stop the ESB. For example that's what the start/stop scripts does (they are in the bin folder). Likewise OSGi has API for controlling bundles as well, so you can start/stop individual bundles. However often people use the shell or web console for that.
To send messages its often easier to use Camel, as it makes it easy from a client to send a message to any kind of endpoint.