why would I want to use that?
It reduces the number of languages your projects use. For a modern webapp, you need developers with some JavaScript knowledge anyway, and if you need only JavaScript expertise, you'll have a bigger labor pool to choose from than if you need JavaScript+Java or JavaScript+C#, or JavaScript+anything.
It seems like it would execute way slower than pretty much any other language,
Can you give any reason why you believe so? Surely this is entirely a matter of VM implementation and thus not something you can make definitive statements about. Besides, it's almost a truism that webapps are usually bottlenecked on DB access rather than CPU.
it also lacks many conventions that more sophisticated languages have.
Um... No? Nowadays, JavaScript (or, properly, ECMAScript) is a quite sophisticated language of its own. It has exceptions, closures, associative arrays... really, all you need.
Is it possible to hand entire objects from the client to the server, manipulate them and return them back?
Not the objects as such, since the underlying protocol is still HTTP, but using JSON is really almost the same thing.