When I use CTRL+H I end up on the Java Search tab. I would very much like a shortcut to go directly to File Search instead. Is that possible?
You can just define a key binding that opens the file search:
You can either re-use the CTRL+H binding (delete the other binding in that case) or define another one (e.g. CTRL+SHIFT+H). To delete the other binding search for "Open Search Dialog" and click on Unbind Command.
Other solution: You could press CTRL+3 in your editor, type in "file s", press Enter. The next time you press CTRL+3 "File Search" is at the top.