Guice is supported on GWT using GIN. For Spring-like DI with GWT, check out GWT Toolbox or Rocket GWT.
I believe GIN is a more natural choice for GWT. Not because it's also made by Google, but because using XML for GWT configuration makes absolutely no sense. Everything gets statically compiled into JavaScript so there is no need for externalized configuration. Keep your refactoring tools happy; go for GIN.
To answer your other question, you will not find many SE frameworks that work on GWT. First and foremost, it has no support for reflection or bytecode manipulation (everything is JavaScript), which immediately rules out a lot of frameworks. Log4j, on the other hand, doesn't make sense because there is no file system accessible on the client side, but there are libraries available that do things differently.
The Spring libraries for GWT mentioned above are basically a rewrite of the Spring for GWT. They do not share any code with Spring simply because they can't. Those frameworks work by generating code ("factories") that wire up your components as if you were doing DI manually.
This is also how GIN works, it generates Java factories for your classes, and GWT compiles it into optimized JavaScript (meaning little performance overhead). GIN does use Guice behind the scenes though, to validate configuration at compile time and to inspect modules.