They're mating.
Dogs stick together when mating to ensure successful reproduction through physical connection and to prevent other animals from interrupting the process.
You cannot
sometimes, it depends, most snakes stick with their "type" but there have been rare cases when snakes interbreed.
On the gear stick right where your thumb rests
Thay are mating or having sex
A male stick insect will not lay eggs. However, a female stick insect is parthenogenetically, and can lay eggs without mating with a male.
The exact meanings depend on what you take the symbols to mean. In symbolism, snakes can refer to both sexuality and healing. Whiteness can symbolize purity, sterility, and/or healing. In the context of mating, this could symbolize safe and hygienic sex. In the context of healing, this could represent complete recovery. The caduceus is a symbol used to represent modern medicine. It is comprised of intertwined snakes on a pole. The origin of this symbol seems to come from an account in the Bible where God cursed people with sickness and they would be healed if they only looked at a serpent on a stick. So the imagery of that story is used to depict the medical profession.
A talking stick, as used by the Native American Indians, is a stick passed from person to person while in a meeting. Only the person who is holding the stick is allowed to speak. If there is another "talking stick", I have not encountered it.
Dogs stick together after mating because it helps ensure successful fertilization of the female's eggs. This behavior, known as "copulatory tie," allows the male's sperm to stay inside the female's reproductive tract for a longer period of time, increasing the chances of conception.
Indian stick insects are the best known of all stick insects. Males are typically smaller than females and are more aggressive, especially during mating.
If your talking about a stick with two snakes wrapped around it that's a symbol at many hospitals, it's called a "caduceus."