Question
And all of a sudden, we were wondering whether or not plants (Any plant, trees, bushes, etc.) can get: cancer, STDs, or possibly the flu. Possible, or not?

Answer
No not the same type as animals. But they get similar diseases.
Cancer
Agrobacterium tumefaciens is a plant pathogen that causes a disease in dicotyledonous (broad-leafed) plants called crown gall. A crown gall is a tumor that usually forms on the stem of plants that consists of rapidly dividing cells. The crown gall is essentially a plant cancer. These galls do not kill the plant directly but they tap resources from the plant and can sometimes grow large enough to block the vascular system of the stem.

Plants are susceptible to a variety of virus diseases, although the general term 揻lu?is not used in the plant world. Fortunately, the virus diseases that infect plants are unique to plants. They don't infect people.
The discovery of a virus-infected plant can be frustrating. Diseased plants cannot be 揷ured,?and the best course of action for gardeners is typically to remove them to avoid spreading the problem. Viruses also can infect important crops such as wheat, rice, soybeans and citrus fruit. The result is often poor yields. There are hundreds of different viruses known, many of which are specific to a certain type of plant.



Since plants don't sneeze, spewing virus-infected particles to their companions around them, plant viruses require a helper, known as a vector, to move from plant to plant.  Vectors are most commonly insects, such as aphids, leafhoppers, thrips, whiteflies and beetles that spread viruses when they move from plant to plant during feeding. Other vectors include mites, fungi and nematodes (microscopic worms). Humans can also spread plant viruses. Pruning, harvesting, grafting and other plant handling chores can potentially transfer virus particles from plant to plant.

As far as Sexually Transmitted Diseases--Plants do produce sexually and some diseases are spread by insects that are required to pollinate the flowers so I guess in a small sense they can get STD but not anything like animals.