Are there species which is debatable if they should be classified as animals or plants?
No, there are no such debates. Plants and animals diverged over a billion years ago. Animals are defined by the unique molecules that are secreted by the cells, and fill the intercellular matrix, like collagen. Plants have unique cell walls, chlorophylls a and b, and plastids. The unique characteristics of plants are given here: http://www.bio.miami.edu/dana/160/160S10_10.html
Although there may be animals like sponges that don't seem to move around a lot, or corals that have symbiotic relationships with algae, simply "behaving like a plant" does not make them plants. Similarly, there are plants that move and trap insects. But taxonomists would never confuse them with animals.