There would be plenty of gravity. Why shouldn't there be? Your plane would need to be travelling at about 8 kps (with the rotation of the earth playing no role! Why should it?) before you would cancel out the Earth's gravitational field.

Your question contains an invalid assumption that gravity is related to rotation.  In classical physiscs gravity is simply an attraction between masses proportional to their magnitude and distance.  Their relative rotations are not significant (although I do recall some unverified research trying to prove hyper-rotating discs reduced the apparent weight of objects above them).

In this case the weight of passengers in your aircraft would not vary whichever direction they flew.  If they wanted to achieve weightlessness the pilot would have to fly a trajectory such that the aircraft was heading towards the Earth's centre of mass at the same speed as a body would free fall in a vacuum.  If you would like to see the calculations then Wiki "vomit comet"

The Earth does not rotate clockwise unless you are above the South Pole looking down on the planet. If you are above the planet looking down on the North pole, it rotates anti- (or counter-) clockwise.  From above the equator, again looking down on the planet, it rotates west to east, because the sun always rises in the East.