Do animals feel emotions, or display empathy?
The answer to the question "Do scientists agree whether animals feel emotions, or display empathy?" is No. However, there are some animals that certainly do not have emotions or empathy (for example animals that have no substantial nervous system, like sponges), and some animals that certainly do have emotions. The only disagreement is in the middle and where to draw the boundaries. In reality, drawing such boundaries is probably not all that productive, coming down to semantic rather than biological argument.

As a citation for the disagreement, see the two answers and the comments here already.

The difficulty in assessing a statement like this is largely one of operational definition: you need to define what an emotion is, and depending on that definition you will get entirely different answers. You will also get different answers depending on your philosophical approach to questions like these and what the essence of experience is. I would also point out that emotion and empathy are vastly different levels of cognitive ability according to typical definitions of the two.

However, I will say that there are a lot of recent findings of higher cognitive abilities in non-human animals than previously thought. Rodents have long been used as generic models of mammalian nervous systems, including emotional processes. Mirror neurons in non-human primates respond to the behaviors and movements of conspecifics, which is an important aspect of both social learning and potentially empathy. Self-recognition is an important aspect of theory of mind, and has been observed in several species of different types. Mourning behaviors have been observed in various animals in response to death of conspecifics. There can be some disagreement of the interpretation of these findings, but I'll include some references here so readers can do some of their own research and find if they are convinced.

Note that these works study particular animal abilities in particular contexts, rather than making blanket judgments about "all animals" or anything like that. This is typically how science works, incrementally based on the results of individual experiments.



References

Gallese, V., &amp; Goldman, A. (1998). Mirror neurons and the simulation theory of mind-reading. Trends in cognitive sciences, 2(12), 493-501.

Lane, R. D., &amp; Nadel, L. (Eds.). (1999). Cognitive neuroscience of emotion. Oxford University Press.

Mulcahy, N. J., &amp; Call, J. (2006). Apes save tools for future use. Science, 312(5776), 1038-1040.

Plotnik, J. M., De Waal, F. B., &amp; Reiss, D. (2006). Self-recognition in an Asian elephant. Proceedings of the National Academy of Sciences, 103(45), 17053-17057.

Prior, H., Schwarz, A., &amp; Güntürkün, O. (2008). Mirror-induced behavior in the magpie (Pica pica): evidence of self-recognition. PLoS biology, 6(8), e202.

Rajala, A. Z., Reininger, K. R., Lancaster, K. M., &amp; Populin, L. C. (2010). Rhesus monkeys (Macaca mulatta) do recognize themselves in the mirror: implications for the evolution of self-recognition. PloS one, 5(9), e12865.

Roth, G., &amp; Dicke, U. (2005). Evolution of the brain and intelligence. Trends in cognitive sciences, 9(5), 250-257.