Yael,
>> What about UK citizens that pay via PAYE? Are they required to file a tax return as well?
There are three circumstances under which you would either need to or want to file. Read my post again. That's your answer.
>>Also, is there a certain date that the UK tax return is due?
Due 31 Jan 08. If you have a refund, obviously file sooner. If you owe, file 31 Dec 07, and then you will have more Foreign Tax Credits if you are using the paid method on your Form 1116 Foreign Tax Credit (on your US return).
>>Do you know the name of the forms I need to submit?
The form is a "Self Assessment" form, or SA, and found on the HMRC website.