All of the above. The feed dogs are made of one solid piece of metal, but they can be slightly out of line. If they are, it's easy enough to notice; they align with the space in the needle plate which is always straight. Fabric feed can misbehave if there's lots of lint packed under the needle plate, a good cleaning often improves odd behavior.
For pulling side ways, I would like Gless suspect the presser foot, but in most cases it is the pice of fabric itself that weighs down on one side more than the other, and drags it out of line. If you test sew two layers of smaller pieces of fabric it might behave differently.