Let's assume that Earth in its current state is suddenly replaced with the Earth completely identical, but made entirely of antimatter. Ignoring meteorites and asteroids which would cause mass destruction, extinction events and whatnot, what would the night sky look like as the anti-Earth swept up the space dust?
It's kind of a strange question but I'll give it a shot.
It depend, kind of obviously, on how much of the "space dust" hits the earth. Estimates vary pretty widely on how much dust from space hits the Earth anyway, by this site, between 5 and 300 metric tons per day.
Lets start with the low end estimate, 5 metric tons (5,000 KG) per day. Antimatter will meet with and turn into energy an equal amount of matter, so 10,000 KG per day turned into energy, presumably almost all gamma-rays, but a lot of that energy would reflect off the upper atmosphere, lets estimate 40% of it reaches the earth, so 4,000 KG of gamma ray energy hits Earth from Space every day.
4,000 KG, * C^2 (300,000,000^2) = 3.6*10^20 joules of energy per day.
The sun (I'll just borrow from this link rather than calculate). 1.6x10^22, so, the heat from the matter-anti matter interactions would be 2% of sunlight. That would be enough to warm the earth measurably. A 2% increase in energy form space could warm the Earth several degrees, maybe 10 degrees, maybe more. The Earth would be measurably warmer, but that would be the least of our problems.
I think we can bring the number down a bit, cause you said no meteors, or asteroids, and even if you hadn't, larger objects like that in a matter-antimatter interaction, the contact would be so hot that most of the meteor wouldn't have a chance to evaporate but would simply be exploded away from the Earth, so we can round the number down quite a bit, like maybe a few hundred KG per day gets converted to gamma-rays and travels through the Earth's atmosphere and hits the Earth. At this stage, the total heat the Earth gets is negligible enough at those levels, but would the sky glow visibly at night? Honestly, I don't know, but maybe.
Gamma rays aren't visible, but as the rays pass through the atmosphere you might get enough visibly hot atmosphere/visible thermal radiation that you might see it. I don't know how to begin to calculate that, but it seems possible. The gamma ray effect would be similar to what would happen to the Earth if it was hit by a gamma ray burst (from a non lethal, but perhaps uncomfortably close distance). Here's a description of what that would be like. We'd lose much of our ozone layer and the chemical composition of our atmosphere would change. over time, it could end much of the life on the Earth's land. Life in the oceans could survive, but life on land would have a hard time.
That's not the only effect though. As a few hundred tons of antimatter hits the upper atmosphere, the upper atmosphere would heat up and the Earth would lose it's atmosphere due to higher temperature much faster. Over thousands, perhaps millions of years the Earth would lose much of it's atmosphere, which wouldn't be fun.
A final effect is that antimatter would change chemistry. If an Oxygen atom is hit it becomes a Nitrogen, if a Nitrogen is hit it becomes a carbon (I think, unless the gamma ray energy of the evaporation causes further photo-disintigration), but you'd see chemical changes as a result, perhaps some of them toxic and some radioactive isotopes. So, no ozone layer, gamma ray radiation and toxic chemicals forming in the upper atmosphere - not exactly sunshine and puppies.
Teeny-Tiny amounts of antimatter hitting the Earth isn't an issue, but in the amount you propose it would probably make the earth pretty inhospitable pretty fast.
I think night sky will be the same as it look today. But if only earth is the only antimatter present in the solar system then it will destroyed after some time and converted into energy. As we know antimatter + matter = energy. But the night sky will be as same as today.