Since Cursive works by analysing source code, for the most part everything that works for Clojure just works for CLJS too - navigation, refactorings etc. I'm planning to add very soon native support for CLJS REPLs, since getting them running can be pretty fiddly. And mid-term I'm planning a CLJS debugger with integrated REPL. That would also allow debugging CLJS in Electron shells or on devices when using React Native.
Can a browser REPL-connected Cursive auto-complete based on the browser environment?
Not based on the live browser environment over the connection, no. Currently the completion for CLJS REPLs comes from the current project, and Cursive uses the Tern static files to autocomplete basic things that are normally present. I'm planning to improve this, but that's the state of play right now.