Is there a debug log or something that would allow me to see what is happening while I wait for queries to start executing?
In your SQL Developer installation's sqldeveloper.conf file, try switching to debug mode ...
Then in your View > Log > Logging Page - Log tab you will find some additional details.
I experience similar behaviour when I'm pressuring SQLDeveloper for memory - lots of open connections and pinned resultsets, editors with huge files, doing schema comparisons etc. When SQLDev memory usage is at about value specified by Xmx option in product.conf file, then JVM garbage collection kicks in before doing anything, resulting in long waits (3 - 10 seconds before sending the query to the database), slow text rendering in editors (letters appear 2-3 seconds after typed), slow window refresh, high CPU usage etc... which makes it effectively unusable. Just enough to save everything and restart the tool.
Try increasing the value in option Xmx in your product.conf file and see how it goes, eg.
Take care to close result sets, unneeded connections etc. Make sure it is not your work habits that push the tool "too far".