Can animals have Down Syndrome?

Or to be more precise, can animals be born with a rouge “extra chromisone” that causes physical defects the same way a 23rd chromisone causes problems for humans?

Yes.

http://www.np.edu.sg/~dept-bio/rdna/geng/downsyndrome.htm

Yes.

http://www.np.edu.sg/~dept-bio/rdna/geng/downsyndrome.htm

Of course, those syndromes are only similar to Downs Syndrome, and not exactly the same, because different species have different chromosomal arrangements. Most trisomies are invariably fatal. Only trisomies of small insignificant chromosomes are likely to be survivable.