We used to joke when I was in college that particle physics was the act of kicking a dog and using the bark emitted to determine everything about the dog, more or less the hammer/nail analogy. But Hawking radiation wouldn't function that way, the more massive a black hole the more gravity it has, more gravitons, HR works the opposite way, very large black holes emit almost none, very small ones emit huge amounts and it goes with the square of mass inverted. Double an objects mass and you'd expect double the gravitons, but 1/4th the Hawking Radiation if it had an event horizon. In any event this gets into 'virtual gravitons' that in theory could ignore the event horizon but the whole Quantum Gravity much like Grand Unified stuff is hazy, troublesome, etc.

Uniformly distributed at a certain scale, once randomness and local effects cancel out. Spill a trillion peas into the Sahara and some will clump together, some areas will be empty, some will form smiley faces, zoom out enough and the effect is even distribution. Somethings do this nicely, gases at room temperature and pressure in a spherical shell without gravity or any external radiation source will be pretty smooth but if you zoom in enough you'll see clumping and patterns. Put it back into gravity and you'll find more particles near the bottom and near the top fewer faster moving ones. Examine a smooth table with a microscopes and it resembles mountain ranges and valleys and they unique, not patterned.

As the mediocrity principle, well we call these things principles because they're unprovable assumptions but ones not contradicted by available evidence which also seem reasonable.

It can radiate away all of it, it is true that if a speck of space dirt absorbs a photon it will pick up momentum equal to the photons, but if another strikes it from the opposite direction it will come back to a relative stop and need to spit out all that energy. It will do it based on its temperature, if its absorbing more than it's emitting its temperature will rise, and it will emit more radiation as a result. Goes with the 4th power of absolute temperature. Which is to say an object sitting a 3 Kelvin, insterstellar void, is 100 times cooler than Earth, 300 K, and thus the Earth emits 100^4 or 100,000,000x as much background radiation as it would if you shut the sun off and let the Earth cool to Universe standard. A red Dwarf, warmed by its internal fusion to 3000K, would give up 10 Billion times as much as if it simply were shut off. Everything in the Universe is being constantly pounded by photons, outside of proximity to a star, maybe 100 AU or so, this evens out so they tend to come about evenly from all directions, though there'd always be some gradient of course, but the random dust mote is being smacked around constantly by photons in good old fashioned Brownian motion. And a lot of them too, space is not very dark at all, some orb with a square meter of surface area sitting the void at 2.7 K will emit σT^4, σ ≈ 5.67 × 10−8 Wm2K4), or 3 microwatts, and because a microwave has only about 10^-23 joules of energy per photon, that means every second it is emitting about 3x10^17 photons, or about 300 million billion of them a second. It's also being hit by the same number a second from basically every single direction so all that momentum it is absorbing neutralizing out and the energy has nowhere else to go but out via photons.

If I shine a light on it, say a 40 watt flashlight, it will begin to warm up (and move away) until it begins radiating 40 watts minus whatever small amount is going into momentum, if I stick to flashlights on it, either side to hold it stable, it will warm up until it gets to 80 watts of radiaiton. In this case 80 = σT^4 --> 10/σ = T^4 = 1.4x10^9 --> T = 193k, or -80 celsius, if I stuck 6 flashlights around it, top/bottom, left/right, front/back it would rise to 254k, or Montana in the Winter. It would take some time to get there, it's true, based on its mass and such, a hollow beachball doing so much faster then a lead ball, but it would reach equilibrium and pretty quickly in most cases. Minutes, days, when talking about objects that aren't big like dust or rocks. A one ton object with a specific heat of 1 (water) stores 4 million joules per degree and most objects are within an order of magnitude of that, so our orb here absorbing a 40 watt flashlight gets there in around 100,000 seconds, just over a day, call it two since its radiating significantly as it warms too, just not as much. Most stuff in the Universe absorbing light is smaller in mass compared to absorption, a big space rock doesn't absorb as much light as an equal mass of gas, so this effect takes a lot less then a day.

Back to our closed solar system though, with an equal amount of sun and gas. Now it depends on the star, but our own emits 4x10^26 watts and has a mass of 2x10^31 kg, and again it takes about 4000 joules per kg to raise things a degree. It's going to take about 200,000,000 seconds for the sun to heat that other gas up a single degree, to get it to its own 5000k it's going to need a trillion seconds. That's only about 300,000 years though. Of course the calculations of how much dust were based on how much was needed to block that much light at a certain radius and its far lower, enough for the heating to occur in mere days. The infinitely reflective surface is imaginary but analogous to Olber's Paradox conditions, there's light coming from everywhere replacing what leaves/

I usually consider science's greatest accomplishments to be penicillin, man on the moon, pacemakers, plentiful free online porn and such but yes a case can be made that uncovering various paradoxes as of yet unresolved is quite an accomplishment. Keep in mind that many of these paradoxes, like Olber's have been solved and aren't paradoxes at all, this may or may not be true of the rest of them and any others we uncover but I don't subscribe to the notion that there's always some new mystery to be solved, just ones we can't solve or haven't solved yet