If there is no atmosphere on a planet that is rotating on an axis and a rocket is launched "Straight up" from its surface, won't the rocket still have an angular/orbital velocity because the planet surface's rotation has imparted it's rotation to it? Also, would that orbital/angular velocity increase the higher rocket goes straight up?

(Kind of like a person walking straight out from the center of a merry-go-round covers more distance in each rotation the further out they walk -- thus the faster they go on the outer arms of the merry-go-round.)

Then also would not the rocket be in Stationary orbit straight up above that same point on the planet?

Please help me to understand this not so "straight up" concept.