How can I make the drop-down arrow in a Web Part menu bar always display in Sharepoint 2010?

You should check your priority of CSS imports and make sure your custom CSS is actually being applied because the code you have is correct. I was able to verify that on a standard page, adding a webpart did not show the drop-down arrow. When I then added in your code (and actually I didn't need the!

Important) then the arrows were shown by default. There was no JavaScript getting in the way, it was all controlled by CSS. So the problem may be somewhere else in your code.

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