What angle does the incident ray from point A on the ground make with the ground so reflected ray off curved mirror reaches the eye above the ground? 
In this illustration:

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*E is the eye of the observer, projecting E straight to the ground gives E'.

*A is a red dot on the ground.

*P is the point of incidence on the mirror. P' is the projection of P on the ground.

*AE makes an angle α (angle EAE') with the ground.

*The incident ray AP makes an angle β with the ground (angle PAP').

*Thinking of the mirror ball as a globe-like object, the circle on the ground whose center is O is the projection of the "latitude" of the ball that goes through P, and that projection goes through P'.

Here are my questions:

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*Is it possible to know angle β? Is it possible that β equals α?

*Does line OP' bisect angle AP'E'?

Here is another illustration, with circle O simply being the bottom of the cylinder. The same questions for this case.

 A: 

May be above two Figures answer a part of your questions.
Note : double-clicking on the images you'll have larger and more clear views.
A: 1)  "Is it possible to know angle β?"
Yes, in principle, but very it's very complicated.  There are two answers, so far, for your two dimensional version of this question here
Given a specific incident ray and a specific reflected ray, how to find the exact point of incidence on a circular convex mirror?
for someone to find a method, they would need to know the radius of the sphere or cylinder, the positions of $A$ and $E$ and they might need to do it in terms of an angle $\gamma$, angle $OAP'$
2) "Is it possible that β equals α?"
Sphere: Yes
Reason: Imagine we increase the height of $E$, $\alpha$ could be very high, but $\beta$ can't go higher than the angle to the top of the sphere.   So for a high $E$, $\alpha \gt \beta$.
If $E$ was very low, $\alpha$ would be low, but $\beta$ can't be too low, the reflected ray would be reflected into the ground too early.  So for a low $E$, $\beta \gt \alpha$.
Therefore it must be possible to find a height for $E$ where $\alpha = \beta$, although in general they would not be equal.
Cylinder:  No
Reason: $AE$ is shorter than $APE$, but both paths would have to do the same change in height so $\alpha \gt \beta$, always, in this case.
3) "Does line OP' bisect angle AP'E'?"
Cylinder: Yes
Reason: If we changed the height of $E$, the height of $P$ would have to vary, but it would still be above $P'$, so it's like replacing the cylinder with a plane mirror.
Sphere:  No
Reason: The above would not be true for a sphere and the position of $P'$ would change, so in general the angle bisection wouldn't occur, although there would be certain positions for $E$ where it could occur.
