get in an elevator

It has no steps!

Not necessarily. An hydraulic elevator (as in mining), or a grain elevator would not be pulleys. Nor elevator shoes.

No. The word "elevator" can be a noun ("Get on the elevator!") or more rarely an adjective ("Check out my new elevator shoes.")

*uses such as elevator shoes or elevator shaft may be considered a noun adjunct rather than an adjective

The elevator ('lift' in the UK).

He did not invent the elevator(lift). He invented the elevator brake system.