Is the converse true as well, i.e., does conservation imply that symmetries exist?

I think this is what Lie groups are all about, but that's a bit deeper in group theory than I'm comfortable speaking on.

I'd be interested if you know any other good resources for learning this.

I learned it the long way by taking classes, and don't recall being particularly impressed by any textbooks. (I can lend you the ones I used.) I remember thinking that reading through Akella's lecture notes was about as good as taking the course, and so if you have the time to devote to it you might be able to get those from him by asking nicely.

Conservation gives a local symmetry but there may not be a global symmetry.

For instance, you can imagine a physical system with no forces at all, so everything is conserved. But there are still some parameters that define the location of the particles. Then the physical system is locally very symmetric, but it may still have some symmetric global structure where the particles are constrained to lie on a surface of nontrivial topology.