We gave dd a pillow at about 26 months...it happened by accident as she had a cold and I wanted to elevate her head a bit to help her breathe. When I tried to remove it, she went bananas....so that was it! My dd has slept with a blanket forever, but her crib-set didn't come with a comforter, but rather a small velvet blanket that she has always used. I know that is a bit controversial, but we did it as she seems to really like being covered rather than not.
You can find toddler pillows that are flat, I'm pretty sure "The Right Start" has them online.