Sure, a handyman can snake out a drain, but he cannot call himself a licensed plumber on his truck or stationary. He also cannot pull a permit in many states.
Journeyman Plumbers are licensed.
And no, snaking a drain doesn’t require a license, although again it’s a state by state matter. The regulations apply to working as a Licensed Plumber, as opposed to a handyman for instance. What states allow will vary, for instance an installer of underground lawn sprinklers may not require a plumber’s license but still might require some other certification.
Every time I’ve looked for a number online as used to be found in the White Pages, there’s a charge to see it. Most services like plumbing you find on non-yellow-pages web sites.
As a side note, the “Joe the Plumber” who complained Obama’s taxes on over-$200,000 income meant he could not make enough money to buy a plumbing company he allegedly worked for - was not a licensed plumber either.
A trap is required, but there are alternatives to a p-trap. I’m not sure if they meet code requirements in all locales.