fix: LSP tool startup reliability, shutdown hangs, and refactor into focused files

1. Server won't restart after a crash. startLocked() checked h.cmd != nil
   to decide if the server was running, but h.cmd stays non-nil after the
   process dies. A crashed server could never be restarted — every subsequent
   call got "LSP server already running". Fixed by adding a processDone
   channel that tracks actual process liveness.

2. stopLocked() hangs forever. It called sendRequestLocked("shutdown")
   synchronously, which blocks on reading stdout. If the server was
   unresponsive or already dead, this blocked indefinitely — and so did
   cmd.Wait() after it. Fixed by running the shutdown handshake in a
   goroutine with a 5-second timeout, then killing the process via context
   cancellation.

3. Data race on the stderr buffer. os/exec writes to a bytes.Buffer from
   one goroutine while readNotifications reads and resets it from another.
   bytes.Buffer is not thread-safe. Fixed by introducing a mutex-protected
   lockedBuffer.

Assisted-By: docker-agent

Author: dgageot