. . . floor.¹

By convention, angles are measured counter-clockwise from the reference axis, so technically $\theta_1$ is negative; this makes no difference to its cosine. Thus $v_x = v \cos \theta$ is initially positive and $v_y = v \sin \theta$ is initially negative.

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. . . negative.²

Again the convention is to treat counter-clockwise rotations as positive; by inspection we can see that $\omega_2$ must be clockwise, so $\omega_2 < 0$. As long as a consistent notation is used, it doesn't matter if the direction shown is opposite to the actual direction; the answer will always come out of any correct calculation with the required minus sign.

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