Question

Can DNA "left over" from genetic engineering infect other organisms?

Answer 1

Short answer is no, the DNA would have no way to penetrate the cell membrane and enter the cell. It would also be broken down by enzymes. That is unless the DNA was heavily modified to overcome the reasons mentioned.

Answer 2

No, this would only be possible if it's carried by a vector (mostly, a virus) that would carry it inside the nucleus. Besides, the DNA fragment would have to possess the right contents to be implantable, which in a random fragment is as likely as a rat's tail getting successfully grafted on a random portion of another species' body.