Zinc doesn't rust; zinc protect iron (and alloys) from rusting.
zinc is used to prevent rust.
Yes, they will rust due to zinc.
Zinc will absolutely corrode. In fact, will actually do so faster than many other metals. This corrosion is not called "rust", because rust can only come from iron. So, zinc does not rust, but it definitely corrodes.
A penny will rust faster because it made entirely of zinc with a thin outer coating of copper, and zinc is made to rust unlike nickel which is used to make nickels, dimes, etc...
Most commonly, the coating on a nail that resists rust is zinc.
It can be protected by galvanising its surface with Zinc.
Protection against rust
Aluminum is not generally galvanized as the zinc would not be readily sacrificial for aluminum. There are steel products which derive rust resistance from a coating of aluminum and zinc alloy. Properly maintained, they will be rust free for 20 years or longer.
Zinc is commonly used to coat iron to prevent rust.
Pennies don't rust. Rust technically speaking is iron oxide and pennies have little to no iron. They do however oxidize, tuning them green from the copper which makes up most of the metal they are made from.
galvanization.
In galvanized steel, a thin coating of zinc is used to protect it from rust or corrosion.