Question: For a fantasy, I need to know how a world similar to Earth would exist in a geocentric model. 

1) I would assume the sun would have to be a lot smaller. I'm okay with artificial stars (hand-wave that stuff away with magic). But I would like to know if that makes an Earth-similar planet impossible. (I don't need other planets in this model.) For example, how would it effect... 

* seasons and climate
* length of day, month, or year
* sunrises and sunsets
* gravity
* constellations and/or navigation
* any huge effect I don't have enough science to anticipate

2) How would I manipulate my universe's model to make it more Earth similar if those things are completely off? It doesn't have to be exact, but I need a temperate climate with pretty normal seasons and climate zones.

Answer: 
The first thing we must understand is that from a purely *kinematic* point of view, the heliocentric and geocentric models are both equally correct within the accuracy limits of astronomical instruments available before the, say, 16th century. There was no way for an astronomer who lived before [Tycho Brahe](https://en.wikipedia.org/wiki/Tycho_Brahe) to bring serious arguments in favor of one or the other. (That's why Galileo Galilei had such trouble with the astronomical establishment of the day -- he simply did not have any good arguments to bring in favor of his pet theory.)

The problem is that from a *dynamic* point of view, for a geocentric model to be correct it is necessary to abolish the law of universal gravitation: and this means, of course, that the universe in which a geocentric model is correct from a physical point of view has vastly different physics from ours. Whether *"a world similar to Earth would exist"* in an universe with vastly different physics than ours is not something that anybody but you can answer.

How would it effect...

* Seasons and climate:

We don't know. The law of universal gravity doesn't work in your world, so we have no idea how wind works, how the water cycle works, the lot. By the way, how does *fire* work in a world where the law of universal gravity does not operate?
* Length of day, month, or year:

Those are purely kinematic phenomena, and from a purely kinematic point of view the heliocentric and geocentric models are both equally correct within the accuracy limits of astronomical instruments available before the Renaissance.
* Sunrises and sunsets:

The sun will rise and the sun will set. We have no idea how the atmosphere works, or how thick it is, because the law of universal gravitation doesn't work in that world. So we don't know if, for example, the sun will appear red at sunset.
* Gravity:

*Our* kind of gravity doesn't work in an universe where the geocentric model is correct. It must be some different force which is called gravity. How it works nobody but you, the author, can say.
* Constellations and/or navigation:

No effect whatsoever. The funny thing is that *up to this day* celestial navigation, as an application of practical astronomy, is done assuming a geocentric model. See [celestial sphere](https://en.wikipedia.org/wiki/Celestial_sphere) for how this works.

Of course, satellite-based navigation systems won't work, because the law of universal gravitation doesn't work.
* Any huge effect I don't have enough science to anticipate:

The main huge effect is that only the author can say how that world works, because it most definitely it doesn't work like ours. What keeps water in the ocean, what keeps people on the ground? Does hot air rise? Why? Are there tides? Why?

Note that you *do not* have to make Sun any smaller or bigger -- whether we adopt a heliocentric or geocentric system has no impact on the distance between the Earth and the Sun.

Everything also applies for the Moon. A Moon may or may not exist; if it exists, it is not universal gravitation which makes it orbit. What is it that keeps the Moon in orbit only the author can decide.
Assuming that Newtonian mechanics still apply the factor that determines whether the Earth goes around the Sun or the Sun around the Earth is the mass of the sun. If the sun radiates equally in all directions then its total power output must be proportional to its distance from the Earth.

Assuming that the mass of the Sun is small compared to that of the Earth, an orbital period of 30Ms (about 1 year) gives a radius of about 2Gm corresponding to a sphere with a surface area of about 5.5E19 m^2. For a power density of 1.361 W/m^2 the Sun must produce 7.4E22 W. If it burns at this rate for about one billion years then it will consume about 2.5E22 kg of matter. For comparison, the mass of the Earth is about 6E24 kg, so if you postulate a light-weight machine with a store of fuel which it converts to energy and radiates in all directions then this arrangement could be plausible.

However, if you have a machine converting fuel to sunlight then why assume that it will waste most of it? If it could focus its entire output on the Earth then it could reduce its power consumption by a factor of about 430,000 and at the same time make the mechanism more easily accessible for the maintenance crew.

Unfortunately the radiation pressure would push the Sun away from the Earth, so perhaps it should radiate an equal amount in the opposite direction. This would produce as a biproduct an interesting galactic lighthouse.

From the perspective of a fairly primitive civilisation this should be almost indistinguishable from a heliocentric system. Weather would be similar, although I am not sure what effect the absence of the Sun's magnetic field would have. If you want a moon like Earth's then you must make arrangements to illuminate it. Maybe spread the beam in the plane of the Moon's orbit, or even provide a separately focused beam. Comets would be interesting, as they would now be orbiting the Earth rather than the sun, and they would abruptly disappear as the passed out of the beam.