Assuming a clear sky and an average flashlight, would any of the photons make it out of the atmosphere or would they all be absorbed? If they did escape, how far could they reasonably travel before getting absorbed?
Well yes, of course, just based on probability. There are an unimaginable number of photons projecting from an average flashlight and some of them will make it past the upper atmosphere into space just fine at about about 186,000 miles per second. The signal from a small flashlight would be very diffuse and hard to detect but still there.
However, the question is unanswerable without specifying the size and type of flashlight. On the extreme end, it is perfectly possible to bounce a laser (a really focused beam) from the Earth to the Moon and has been done regularly. You won’t get the same results with a flashlight from the dollar store but the chances are extremely likely that you can hit the moon yourself with one even if it is a pointless exercise.
For the most part … the atmosphere is transparent to the visible wavelengths of light … so quite a few of the photons would escape into space …
Specifically they bounce off the corner reflectors left there by Apollo missions. And they have to use powerful lasers and very sensitive detectors.
But a lunar bounce is much more difficult than what the OP is asking about. A laser aimed at the moon will spread out over several square kilometers and only a small amount of that will hit the retroreflector. And then it reflects at least twice in that and the small beam from that spreads out on the way back to Earth. As Wikipedia says
What an incredible coincidence! 
In clear, dark conditions, you can literally see a match being lit from fifty miles away. The vast majority of the atmosphere is below fifty miles, and a typical flashlight is much brighter than a match.
No one’s answered the part of the OP about how far the light will travel. The answer is most of it will travel indefinitely. Even if it’s aimed at a nearby object like the moon or Venus, the beam from a flashlight will spread out so much that most will miss. And then keep going until it hits something else, which most of it will never do.
Light pollution …
And then there’s the peak of the solar emission spectrum - visible wavelengths!
Just think of that poor photon, drifting along for zero seconds, wondering when it will ever be absorbed…
I like to think about how a photon emitted in the core of the Sun, traveling at the speed of light, getting absorbed snd emitted many times a second, takes ten thousand years before I see it. But I’m of the Just Say Yes generation.
So when the ISS is above the night side of the Earth (let’s assume the station is within the Earth’s umbra, so local glare is also minimised), if someone on the ground shines a bright flashlight (one of those ‘tactical’ models) at it, and assuming their aim is good, would a person on the ISS be able to see this as a pinpoint of perceptible light, with the naked eye?
In theory, yes. In practice, there’s lots of other brighter lights that it will be in competition with. You’ve seen those pictures of the Earth at night, the ones where you can pick out some geographic borders, such as those of the Great Lakes, from all the city lights around them. Well, your flashlight will be much dimmer than any of those lights, even the smaller dots. Those dots represent the lights from an entire city, albeit one of the smaller ones. If you happen to be in or near a city, even a small one, your flashlight will just blend in, and not really be noticeable from the station.
So if you want an astronaut to be certain they see your light, it would be best to get far away from any city. Out in the ocean for example, although there are also ships and planes there which it will be in competiton with.
I did a Google on % of Sun’s rays absorbed by the atmosphere and got from NASA:
“About 29 percent of the solar energy that arrives at the top of the atmosphere is reflected back to space by clouds, atmospheric particles, or bright ground surfaces like sea ice and snow. This energy plays no role in Earth’s climate system. About 23 percent of incoming solar energy is absorbed in the atmosphere by water vapor, dust, and ozone, and 48 percent passes through the atmosphere and is absorbed by the surface.”
Given that flashlight’s going the other way, far more than 50% of the flashlight’s photons are going to make it out. On a clear, low humidity day away from pollution it sounds like it could start to approach 70%.
You can see the stars right? You know then that light can travel at least that far. 
In addition to the point already made, there’s the issue of angular resolution.
Metaphorically speaking, from ISS orbital altitude each “pixel” in your eye maps to a large area on the ground. WAG a square km or more. And your eye has a minimum brightness it’s able to detect. Which minimum is bigger than one photon per “pixel”. So even if flashlight photons are hitting that one “pixel” in your eye, there may not be enough.
Another issue …
Atmospheric distortion will tend to twinkle the photon stream. Even if the flashlight was magically aimed exactly at your eye’s position on the ISS corrected for photon time of flight, the stream of photons will be buffeted all over the place. Imagine using a garden hose with a high pressure narrow nozzle on a very blustery day. Even with the nozzle stationary you’ll see the stream wobbling randomly up/down & left/right. That’s the effect the atmosphere has on the flashlight beam.
OTOH, there *is *a solution to the too-few-photons problem. Laser Pointer 
“How far that little candle throws its beams! So shines a good deed in a naughty world.”
— Shakespeare (The Merchant of Venice)
Not sure what you mean by that, but it doesn’t seem relevant to this question. Extra photons coming from other terrestrial sources aren’t going to interfere with the flashlight’s photons’ trip through the atmosphere, although they might make them harder to detect, if anyone were trying to detect them (although that wasn’t part of the OP’s question).
–Mark
FWIW, I asked a similar question 5 years ago: