- If the sphere was simply rotated on a single axis, would gravity be constant everywhere on the internal surface? Or would there be areas with stronger gravity and areas with weaker gravity?
- Is there any way to rotate the sphere such that gravity is constant everywhere in the sphere, at least to the point that humans cannot recognize a significant difference in the gravity?
if it is a sphere then no, at the "poles" the apparent gravity would be lower than at the "equator" this is linear so if you half the distance to the axis then the apparent gravity is halved. Near the poles gravity also would be a bit sideways but that is easy to solve by a terrace-like layout.
You can solve this by making the sphere a cylinder (or stretching the sphere to a more oblong shape) where gravity would be consistent on the surface. Or by rotating different latitudes at different speeds.
There are a variety of different ways in which artificial gravity (the experienced acceleration) would not be constant inside a rotating sphere. There is no way to reduce these discrepancies sufficiently to make them indistinguishable to humans (apart from the unhelpful trivial case of slowing the rotation and making the gravity everywhere so low that humans cannot perceive it).
Variation in strength from equator to pole
Standing on the inside of the sphere, you are moving in a circle around the axis of rotation. This circle is largest at the equator, and gets gradually smaller until it is zero radius at the poles. Since the time taken for one revolution is the same everywhere, there is maximum gravity at the equator, and zero gravity at the poles.
Variation in slope from equator to pole
The direction of gravity is directly away from the axis of rotation. This means that the surface appears to be horizontal at the equator, but as you walk towards the pole it becomes steeper, much like walking up the inside of a bowl.
Variation in strength when walking or running
Stood on the inside surface of the sphere you feel stationary, but you are actually moving in a circle. If you walk or run in the opposite direction to this motion then you will weigh less. If you move in the same direction as this motion then you will weigh more. If the radius is not sufficiently large, running could make you weightless so that you lose contact with the floor and drift through the air. Increasing the radius increases the speed you would need to attain in order to become weightless, so that running is no longer a problem, but even a large sphere will still have this problem as you approach the poles and the radius decreases.
This will be more of a problem if there is powered transport at greater than running speed. If the outer radius (equatorial radius) is sufficient to prevent problems at running speed, there could still be problems near the poles or on high floors of buildings.
Variation with height
The nearer you get to the axis of rotation (which appears to be above you), the less you weigh. This means that when standing upright your head experiences lower gravity than your feet. To avoid this effect being large enough to cause discomfort, the radius must be large enough that gravity changes slowly with height. This is a problem in a sphere, since the radius of rotation decreases as you approach the poles.
This also means someone in a building a few floors up from you will feel lighter than you, heading towards weightless at the axis of rotation.
The floor tilts when you stand up
The Coriolis effect causes what feels like a horizontal acceleration when moving towards or away from the axis of rotation (that is, during the process of sitting down or standing up, or when going up or down stairs or in an elevator). This combines with the apparent downward acceleration to produce an apparent acceleration that is no longer directly downwards. The sensation is that the floor is no longer horizontal, as if it has suddenly tilted.
For this reason elevators may be allowed to swing so that the floor can align with the temporarily tilted horizontal plane during ascent and descent, returning to normal when they stop.
Again, this effect is reduced by increasing the radius.
Comparison with a cylinder
In a cylinder, you can simply make the radius large enough to make these effects too small to cause problems. In a sphere, a large proportion of the surface area is at a smaller radius of rotation, approaching zero radius as you near the poles, so it is harder to avoid these problems unless you only use the surface nearer the equator.
Even if the outer appearance of the structure is spherical, it would be more practical for any floors inside to be cylindrical, so that the floor appears horizontal wherever you stand.
Even then, people on sufficiently high floors (nearer the axis of rotation) will suffer discomfort and disorientation, accompanied in many cases by nausea. These areas might be restricted to scientific use or storage (or perhaps be very low rent if your sphere has an economy).
This answer is already long so I haven't included specific numbers. If you wanted to know something in particular, like the radius or rotation speed at which running would cause weightlessness, or the change in weight per floor if the outer floor is normal Earth gravity and the radius is a mile, then they could be asked as separate questions.
Rotationally simulated gravity always points directly away from the axis of rotation, and there's no way to change that. So unfortunately the answer is no to the second question.
For the first question, essentially the further you move from the axis then the stronger the simulated gravity becomes. You have free fall right at the axis, then as you move further away the force becomes stronger and stronger.