If you put a piece of "worthless' DNA next to a gene or region of the genome of economic interest in an animal, the DNA of this animal is modified. Would it be possible to get a patent on this animal, and the offspring?

Identifying a gene sequence freely occurring in nature (not in genetically modified animals) is a discovery and not an invention. An animal carrying this gene sequence can not be patented, just because someone can identify the gene sequence or indicate what the function of a gene sequence is, nor can his/her offspring.

Isolating a gene or a piece of DNA, and determining its utility, e.g. developing a selection test based on the sequence information may be an invention and can be patented. However, animals selected using DNA markers or similar tools cannot be patented, as they are produced by normal agricultural practices (e.g. mating or artificial insemination resulting in crossing of genomes).