You're asking lots of things here, and you have to separate concerns. For example, take air (and drag) out of the picture. You can take off vertically from a location on the equator, with a certain speed, but you will not remain above the launch pad, because to do that you would have to accelerate to the east as you go up.
This is because the radial you took off on moves faster as the distance from the center of the earth increases.
Then coming down you would have to accelerate to the west if you wanted to stay on the radial.
If you did neither your travel would look something like this (if you were looking down from the North Star):
You launch "straight up", but you're not actually traveling straight up, because your launch pad is traveling to the east at about 1600 km/h.
So you are actually launching at an angle, and you go into an elliptical orbit.
That orbit intersects the surface of the earth, so that's where you land.
The launch pad gets to the landing point before you do, because it has less distance to travel and because you slow down on the outer part of the curve.
That means you will land west of the launch point.
Notice how this looks to an observer who stays at the launch pad.
You took off straight up, but then your path simply bent to the west.
Then you reached maximum height, still traveling to the west.
Then as you fell down, your westward speed decreased until you landed "straight down" at the landing point, to the west of the launch pad.
Now what about air? There are winds aloft, and you can get them from aviation forecasts.