That's definitely not default behaviour... you should see a "You have chosen to open x. What should Firefox do with this file?" prompt.

I was replying to the quote about the browser asking where to download the file. By default you get that prompt you mention but it won't let you choose where, if you choose to download it goes without asking to the downloads folder unless you've changed that setting in the options page to always ask where. Chrome is definitely a worse offender, it does not even ask you to confirm the download.