The trickiest part about having time zones on a diskworld is explaining why the light doesn't just spread across the whole disk as soon as the sun peeks above the horizon.
Extreme refraction of the light in the atmosphere (or at least in the bubble which keeps the atmosphere on the disk and even many levels of refraction from multiple layers in the atmosphere) could explain this - the light hits the world edge-on and the direction of travel sharply bends diskward. The edge closest to sunrise gets sunlight right away, but the sun has to get high enough for the angles to work out. This would imply the potential for the poles (ends of the axis perpendicular to the sun's path) to be forever dark, as the sunlight might not get that far out, and the night will always be a little longer than the day.
The moon would have to glow with its own light, as the disk will block the sunlight from reflecting off of it at night. This is ok, because it is a magical glow, thus justifying the fantasy concept that moonlight has special powers different than sunlight. Phases of the moon will be trickier - a crescent shape isn't likely, so you could go with an always full moon, but I like the other suggestion for a spinning disk-shaped moon for phases (even get 2 different sides of moon to play with).
Seasons depend on what you mean by seasons - different parts of the planet have different kinds of seasons. The arctic has periods of no sun and periods of no dark, equatorial regions have wet-dry seasons, etc.
Options for temperate warm/cold seasons could be the sun rotating its path - rises in the 'north' at the beginning of the 'year' (very arbitrary notions which could be completely different depending on which culture on the disk you are in), slowly migrating clockwise throughout the 360 degree/day year. The sunrise/sunset areas are warmed by the proximity of the sun, while the bits halfway around the edge are colder (the rock of the disk over there hasn't been heated in a while). The hub wouldn't experience any seasonal changes though, so either always winter (never gets close to the sun) or always comfortable (always gets maximum sunlight), depending on your preference.
Alternately the orbit could expand/shrink - the wider the orbit, the less the warming effect of the sun. This would change the length of the days so probably not a good solution - not like the real world where there is a tradeoff between day and night, but both getting longer/shorter together.
This next one is going to be a little tricky, so bear with me.
Gravity - I'm assuming it just magically pulls 'down', else the edges of the disk would be like climbing a cliff as you are pulled toward the center of mass. This has implications for the atmosphere and any oceans (sail off the edge?), as where does the stuff go that falls off the edge?
If we imagine the center of the disk being a hole to the other side, it could be a font returning all the air and water. The air return is usually just air, but the water accumulates on the 'bottom' of the disk until it reaches a critical level of buildup, at which point it overflows and 'spills' through the hole to be ejected on the inhabited side (like a geyser of mist into the upper atmosphere).
This air could be cold, bringing winter snows, or it could be hot, bringing the rainy season of monsoons.
This would also imply the wind is always blowing out of the hub, but you could go with a whole smattering of different vents which are active at different times (changes the wind directions, and allows for different areas to have different seasons).