My experience with labs is that, no matter how the image is printed, it's trimmed to the image size unless otherwise requested. (I often take advantage of this by submitting odd sized panoramas [ie 5x14 on an 11x14] and the lab trims off the excess)
The frame will cut in 1/8" to 1/4" depending on the frame on each side so that the image doesn't just fall through the window.
A print is often too flimsy to hold itself up, so a cheap way to go is to add cardboard behind the print.
Why you often see prints using a mat with a border is because its fairly cheap and easy to do at home and it looks good. The print can be secured to the front mat using tape on the backside (or the back mat for that matter), and then can be sandwitched between the front mat and a solid mat to give it rigidity in the frame. This option doesn't explicitly attach the whole image to anything though (just the corners), and is akin to cardboard method above.
If you have a large print or something you care about lasting as long as possible while in a frame, your best bet is to mount the print onto a board or foamcore using an adhesive. This gives the print structural support and will allow you to add the print into a frame without any additional backing - though again, cardboard is usually used to fill any extra space behind the print.
Most semi-pro and pro printers will offer mounting services. You can purchase the required supplies, but I would highly recommend practicing with cheap 8x10's until you get the hang of it.