Vented cooler=one with a drain plug. Stroform coolers are a no go container
wet ice=regular ice you use for drinks
dry ice=you are allowed up to 5 lbs per container. This is considered a
dangerous goods and must be declared. Dry ice sucks the oxygen from the
cargo area therefore anything containing dry ice can not be placed near
live animals. It can also burn on contact. CO does not mind you using it
but we must know about it and you will need to fill out some paperwork.

Because you will not want to be liable for any spills, place a plastic wrap inside your cooler to contain any possible leaks/spills. All the ideas for how to keep your items cold/frozen are good ones.