I am currently constructing a fantasy world, where the sun and moon revolve around the planet, which itself is stationary. Because of this, it is quite difficult to imagine how the climate zones would differ from our Earth (where would it be naturally colder and warmer in the geography). I am looking for a program (app, website, software, whatever) which would let me create 3D images and could have simulated light sources, that could help me determine how much sunlight would reach each continent, island, plot of land.
This is a very complicated subject
And I'm not sure anyone has written a program to do everything you asked for. Bits and pieces have been written and polished but there's no cohesive project that does everything you're asking for.
If you have a PC with Steam installed, you can use Universe Sandbox to setup as many strange orbital configurations as you desire. There's some climate modeling done but probably not to the fidelity that you are looking for.
This is probably your best bet for covering lots of ground in a single package. I'd start here.
The MIT Global Climate Model (MITgcm) will allow you to do the kind of simulation you require with really high fidelity to real life. However, MITgcm is real scientific software thus making it unapproachable to anyone who doesn't have a degree in atmospheric sciences.
Having played around with MITgcm, I've learned that climate, atmospheric, and oceanographic modeling with real fidelity is very complicated and not for the faint of heart.
Blender is an open source 3D modeling program. It will only allow you to build a 3D model of a planet. You can animate it from there to get the views you want.
Spreadsheets are your friend. I've written spreadsheets for myself to figure out the hours of daylight for a planet that's tilted over 90 degrees.
Do a bunch of research into climate models, biomes, the relationship of water availability and temperature. As you get more familiar with these subjects, you'll be able to look at a map, figure out some ocean currents and prevailing winds then derive the overall climate for that location.
- Atmosphere Circulation on Exoplanets This paper is wonderful.
- Arxiv.org This place has tons and tons and tons of papers on many subjects of interest to you.
- Climate Zones and Koppen Climate Classification
Climate is very complicated and requires a lot of background knowledge to get right. Start reading everything you can on the subject. Good luck! It's a really fun area to go exploring.
Universe Sandbox might help. It's basically a solar system simulator, on steroids :)
It'd help you establish the orbits and positions of the bodies and the light patterns of the day/night cycles. It's not a planet simulator as such, so I don't think you can drill down into too much detail on the planetary conditions, it'll report and simulate things like temperature and gravity but it'll treat the entire planet equally, I don't think you can break it down into different climate zones.
Adding to the already existing answers and seconding Universe Sandbox which is amazing:
PlaSim is another tool for scientific-level climate simulation at home. However it requires good meteorological knowledge, work, and patience if you want to tweak it to your desires.
Actually I think the answer is simpler than you may think. If the sun and moon revolve around the earth the effect is EXACTLY THE SAME as it is currerntly - the moon has the same tidal effects as it does now, and the sun has the same effect as it does now but at a greater distance.
Thinks about it this way. Assume that the earth stopped rotating around the sun (and didn't fall in because gravity got fired) -- but it kept rotating AROUND ITSELF.
From the earth peoples point of view they experience exactly what you said -- the sun and the moon orbit around us. So, the solar coverage is the same and the effects on terrain are the same.
The main difference is that if the sun stays on the same plane, there are no seasons. Assuming the suns' orbit is on the same plane all the time, part of the earth gets a lot of sun, and part of it gets less.
There might be an accumulative effect -- a lot of scorched earth, a lot of freezing earth, and a relatively small temperate band of livable terrain.
The stars in your scenario remain fixed. Thats irrelevant to climate but its a thing.