How can I exclude a folder from indexing in Sublime Text, while still showing it in the sidebar?

To exclude files from the index but keep them in the sidebar, use the binary_file_patterns setting in your User Settings, for example:

"binary_file_patterns": [
  "*.jpg", "*.jpeg", "*.png", "*.gif", "*.ttf", "*.tga", "*.dds",
  "*.ico", "*.eot", "*.pdf", "*.swf", "*.jar", "*.zip",
  "node_modules/**",
  "bower_components/**"
]

Make sure to copy the values from your Settings - Default preferences (here shown as "*.jpg" etc.), or you will start indexing binary files.