Without access to the proper resources I can't say too much about the Sun and Moon. What I do know is that the Moon and Sun are flat disks about 32 miles wide at an altitude of about 3000 miles. This can be measured with a sextant and calculated with plane trigonometry. There's also a 3rd disk known as the Black Sun, it can't be seen with the naked eye except when it causes an eclipse.
Moon light is coherent and produces a laser-like cooling effect while the Sun's light is obviously hot. The disk layout of the Sun and Moon is what causes the spotlight effect.
We're at the limits of my knowledge here but I'm going to go our on limb and say the Sun and Moon are most likely synthetic diamond disks that are held in place via superconducting levitation. Originally I thought they were projected onto directly onto the dome wall but I feel they're probably separate objects.
Any idea how they're powered? You realize that the Sun radiates 173,000 terawatts of power onto the Earth (this can be tested empirically by anyone using PV panels etc.). That's a fuckton of power, it must be a very heavy-duty generator that's being used. Maybe it's outside the dome, because we obviously can't see it...
Any thoughts on the night-time star projector?