This is the case for mobile phones. On a desktop or laptop, the dropdowns are preferable because most browsers will let you select the day, month year using your keyboard.
Yes. And please - if you feel like reinventing the dropdown be decent enough to re-implement that feature.
Example of an offender: https://www.google.com/advanced_search
To be fair, that page predates Luke's stint at Google by a long time (I don't know how long has it been since it last received an update, many of Google's features are sadly neglected).
I don't have firefox, unfortunately. Anyways, the feature I'm talking about at least (dunno about my parent) is being able to select an item by typing characters - i.e. typing e-n-g should get you English.
You're right that people should be able to type in a date, but I'd argue that it's preferable to have a text input with a button to open a calendar control - many users don't know that you can use the keyboard to select dropdown box items so they'll use their mouse, and that negatively impacts their experience. If you want a user to be able to use their keyboard then the input should look like a keyboard input box.
Do you think I can guess where "United States" is in your country dropdown?
Oh, it's the sixth country, even though everything else is alphabetic order, but Sweden is on top because apparently you're based in Sweden and so is Denmark and Norway because apparently you have a lot of customers there.
But I'm supposed to know this while looking under U and seeing "United States minor islands"? It's really annoying, I just type United and it usually jumps to the correct thing (the first United it sees)
For desktop, I generally prefer checkboxes. Instead of having to discover the available options in dropbox, you can see all of them instantly in a list.