Can life exist on planets as barren as Mercury? How would it survive? It seems like it couldn't. Mercury is very hot, would any life break down becuase of the heat? What other challenges face life on Mercury?
Anything is possible
Life as we know it on Earth could not exist on Mercury without serious terraforming. Mercury has very large temperature swings, the side facing the sun can reach temperatures of around 900 C, while the dark side can drop to temperatures similar to that of deep space. Because of this, liquid water cannot exist on Mercury, and water is a hard requirement of life on Earth.
Mercury also faces frequent asteroid impacts. Because Mercury lacks an atmosphere, these impacts would be unimpeded and highly dangerous. All life would likely be destroyed by them. I would like to note that Mercury does not face bad impacts from solar wind, because it has a strong magnetic field, slightly stronger than Earth's.
The question does ask how life could exist, and my favorite phrase on this site (see top of post) definitely applies. Life as we know it could not exist on Mercury, at least not without impressive terraforming (see next section). But life on Mercury could exist if it didn't rely on water, and could stand rapid and violent heat swings. Or if life could survive in the magma of a planet, then it could live on Mercury. In an Earth based context, these don't make sense, but using soft-science it would be possible.
Another possibility would be to terraform Mercury. The first step in terraforming any atmosphereless planet is to give it an atmosphere. On Mercury an atmosphere would be difficult to maintain because it has such small mass, about have the size of Mars or 0.55 Earths. Gravity enhancers may be necessary to maintain an atmosphere, this has not been tested so we can't know fully. A magnetic field is not a problem, Mercury already has one.
After adding an atmosphere, the next step would be to stabilize the temperature. This would require moving energy from the warm side of the planet to the cold side. Once an atmosphere is in place, it is not necessary to move as much energy as some energy will be kept. Once the temperature is around the planet, most of the energy should be reflected. Then water needs to be introduced along with terrestrial life.
It is possible that underground habitats would be easier to manage, but they would still require significant ability to manage heat. An atmosphere, at least in the habitable space, would also be needed.
It would not be possible for Earth-like life to exist there. Perhaps some kind of metallic-based organism, or something similar to a biological nanobot (an advanced type of virus, maybe?) with extreme self-repair capabilities, extreme resistance to heat and radiation, might be able to survive, especially in areas like the sides of ancient craters, where the sun never fully reaches but which aren't in complete darkness either, and which are less likely to be affected as much by new bombardment.
Mercury is not as hot all over as it looks. There is water ice on its surface that may well be permanent. Granted, that's not necessarily permanent on the timescale required for life to develop.
However, like the Moon, the reason that there's water ice in shaded areas is because there's no significant atmosphere to actively transfer heat. And no atmosphere means no pressure, which means no liquid water (it sublimes from solid to vapour). We don't really have the first clue what something might look like that exists in the absence of both liquids and gases, but that we'd choose to call "life" once we saw it.
So things aren't looking good for native life on Mercury, but it's not quite so simple as "too hot".
If you're talking non-native life, then building some kind of station in a shady spot on Mercury isn't absurd. Underground would probably make more sense, though. But that's just a special case of, "with enough technology you can build a climate-controlled habitat almost anywhere".
It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination
Life is a strange and varied thing... And I guess "Life as we know it..." is a somewhat relative term. New species are still being discovered on a fairly regular basis.
The hypothesis of panspermia would seem to suggest that finding these tiny survivors on inhospitable planets would be entirely possible.
Panspermia is a hypothesis proposing that microscopic life forms that can survive the effects of space, such as extremophiles, become trapped in debris that is ejected into space after collisions between planets and small Solar System bodies that harbor life. Some organisms may travel dormant for an extended amount of time before colliding randomly with other planets or intermingling with protoplanetary disks. If met with ideal conditions on a new planet's surfaces, the organisms become active and the process of evolution begins.
I disagree on the impossibility of life. There are areas on Mercury that experience a milder climate (other than the lack of an atmosphere) than the Earth itself. I have even seen a proposal to colonize them.
The closer you get to the poles the cooler it gets, there's a ring around each pole that averages what we consider comfortable and once you go deep enough underground it stays this way all the time.
While it's not likely that live managed to evolve in such an environment we can't rule it out.
Imagine Mercury with more water. This would create an atmosphere of water vapor. If there was a lot of water the planet would get a thick water atmosphere and have a runaway greenhouse and turn into a steamball (see here: https://planetplanet.net/2016/04/06/no-livable-planets-without-life/). With a modest amount, water would likely simply be transported to cold traps and freeze out, like in craters or at the poles or on the night side of the planet. Since the planet is in 3:2 spin-orbit resonance, its day is about 59 days (its orbit is 88 days).
I can imagine a really interesting setup. There are 3 source of water on a wetter (but not too wet) Mercury: 1) randomly-located ice deposits in large craters; 2) permanent cold traps at the poles; and 3) a cold trap on the night side that migrates across the planet as the planet rotates. I'm not sure whether this migrating cold trap would persist but it's an interesting idea. It would be sort of like an Eyeball planet (see https://planetplanet.net/2014/10/07/real-life-sci-fi-world-2-the-hot-eyeball-planet/) but one that is constantly changing as the planet slowly spins.
Life on such a planet? Well, where there is water on Earth there is life. So, I think it's totally plausible to imagine life in two places: at ice deposits (permanent or migrating/temporary) and in the subsurface, also in water-rich areas.