SAJP,
Good, you know the challenges of the planned endeavor.
Where you get into some trouble is in your insistence that since You don't know how to do it, nobody does.
Th technology necessary to set a sustainable colony on Mars has been with us since the late 1970's. True, doing it would have practically bankrupted the United States, so it wasn't practical. It was however possible.
Being impractical is why nobody attempted it.
The minimum size for a '70's level of technology would have required a population of several hundred thousand people. The minimum population size comes from the necessity of producing everything locally. It takes a; lot of skilled workers to provide the necessary production.
For a colony on Mars, they have to have a place to live and get all the required materials to continue to live and even thrive. Those requirements are 1) air; 2) water; 3) shelter; 4) heat; 5) power; 6) communications; 7) companionship; 8) tools sufficient to their needs and 9) storage.
For the trip to Mars, there are other requirements as well. On the interplanetary trip there is need for supply and shelter mostly. The shelter would have to include some form of radiation protection, as well as providing adequate air and temperature control along with a source of power.
Radiation protection can be provided by 15 cm (6 inches) of water. That is well established, and will give a protection roughly equivalent to a person living in Denver Colorado. NASA has some scientific articles on this going back to the 1990's. Some even earlier. NASA is currently working on a shielding means that seems to consist of something like styrofoam that would need around one to two inches (3-5cm) to provide the same protection. It's being built into the new Orion Capsules. Anything that provides lots of bound hydrogen turns out to shield against most radiation.
This still leaves the proposed colonists with some radiation exposure. The effects of low level radiation are vastly overblown. The truth is that you and I are subjected to constant low level radiation. X-ray, gamma ray, proton, electron and neutron radiation strike us all the time, as do Muons from cosmic ray collisions in the upper atmosphere. but we have systems that counteract it in our bodies. Only however so long as it stays below a threshold level that seems to be around twice the worst case background level. After all, uranium salts are the eight most common salt in seawater, and have been for billions of years. Earth has always been slightly radioactive. Mars is likely to be the same.
Gravity might also be a problem there. So many proposals want to generate it artificially during the trip. We believe that we know how, but nobody has ever done it in space. They have on the ground.
Then, once the proposed colonists arrive, the work begins.
1) The first requirement is air. As you noted, breathable air does not seem to exist on Mars. The atmosphere is over 90% CO2, with around six percent nitrogen and two to three percent water vapor. Trace compounds include most of the Noble Gasses, some carbon monoxide, some methane and even some oxygen. A great many other gasses are also there in PPM (parts per million) or PPB (per billion) quantities. But it is really just impure CO2.
For the trip to Mars and for perhaps the first month on Mars, oxygen brought from Earth would provide most of the air breathed by the colonists.
However, after the first set up is completed, the colony would begin to produce air. That's not really all that hard.
Plants take in CO2 and give off oxygen. It 's a byproduct of their internal sugar production. That sugar then makes the other carbohydrates and proteins we eat. So at least some of the work needed to make air to breath will also make the food to eat.
There are also chemical processes that give off oxygen. The famous red color of Mars comes from iron oxide, rust. Refining iron or steel will if done correctly produce far more oxygen than a colony would need for air.
so requirement #1 can be easily met.
2) Water. Water ice has been found from orbit across most of the face of Mars. It's everywhere, almost. Liquid water can't exist on the surface for more than short times. It's below the sublimation threshold, the triple point. fortunately though, Mars is as you noted, cold. It's cold enough that ice is stable, even on the surface for weeks at a time, and if out of the sunlight practically forever.
So ice will be mined. Heat the ice and you get pure water. Water for drinking, water for agriculture, water for fuel production, water for industrial use. We use far more water for industry here on earth than we do for drinking. They will too.
So insurmountable problem #2 is easily solved.
3) Shelter. The colonists will need a place to live that keeps the air in and the radiation both from the sun and from deep space out. It will also have to provide insulation against the cold of the surface and have ways to move air and water and power and data around inside.
We know how to build those. A nuclear submarine holds around a hundred men for months at a time in a can under hundreds of meters of seawater, with no access to the air outside and no water outside that can be safely drunk. Nobody thinks that anything amazing. So why is the same thing pronounced impossible in space or on Mars? Now submarines are built of strong steel, but three feet of stone is as strong as a half inch of steel. It has better insulating properties too.
Pile up rock dust or anything else that's handy, apply some pressure, add some electricity, and it becomes a sandstone like rock. More electricity and it makes some fine pottery as well. 3D printing stone isn't really a new or impossible thing. Oh, and plastics can be made from CO2 and water. So there will also be 3d printed plastic parts.
Of course, it will be steel re-inforced, just as large buildings here on Earth are. So why is it impossible to do on Mars what we nearly always do here on Earth?
Insurmountable problem #3 is easily solved.
4) heat. This is a byproduct of 5) power. The colony will probably have a bigger problem getting rid of excess heat than it will have of freezing. Power. the two systems being proposed are nuclear and solar. Wind isn't very good, nor is hydro or burning coal or oil. Nuclear works no matter where you are. You just have to get rid of a lot of heat. See #4 above. Solar works about as well on Mars as it does on Earth. Mars is further from the sun. So by inverse square, Mars gets roughly half the solar radation that Earth does. But earth loses half of the incoming solar radiation to atmosphere. So it's close to the same overall.
6) Communications. Radio. I need say no more.
7) Companionship. Sociologists say that the colony would need a population of over 160 people to provide enough of a social network to be stable. I don't think anyone is proposing a colony smaller than that.
8) Tools adequate to the job. This is probably the most questionable right now. Tools however can be made and most likely will be made that can do anything needed. An electric bobcat driven by remote control would be a fine mining tool, I'm thinking. Ditto a forklift. Don't underestimate the abilties of a pocket knife either. Tools will change, but some won't change and won't need to.
9) Storage. Any colony will have to store extra items produced at one time for use at another. Why not? We do it rather a lot here on Earth. But storage is really just a few more buildings. Some will have to be pressurized, some will not. but like any city, there will be storage yards, depots and so forth.
So there, all of Mr. Musks dreams are possible things and a Mars Colony is possible. It would even after several years be a comfortable place. Like living in a shopping mall or a resort hotel.
Or a factory floor.
Most likely all of them in different parts of the colony. A nice comfortable place.
But not a cheap one.
Against all this is the fact that a colony, to be stable has to provide enough value to pay for itself. Mars has a lot, but it has nothing that Earth doesn't have more of. Nothing or very little that is.
Mars can provide a low atmosphere environment for electronics production, but not as good as the Moon can. Such things are currently done in Fab Plants that use things like pumps to get the air pressure down to near zero when they need that.
Though fab plants now are getting so expensive that it just might be cheaper to build then on the moon or some other body.
Mars can provide fuel to spacecraft bound for further destination. Yes, if there is demand. Currently there isn't.
For industrial activity, well, it takes from nine months to two and a half years to get from Mars to Earth or the other way around. That's a long shipping lag. Whatever you ship better be valuable.
There is another model that we don't like.
One way to pay for a colony was done by the British in Georgia and later in South Africa and Australia.
Ship out criminals and 'surplus poor' or indigent people you don't want to have to deal with. Once there, they work their way out of Prison, then have to live in the place they were sent. The process works, sort of. Though the people thus cast off tend to hold resentments for rather a long time.
So there you have it. Colonizing Mars can be done, but not profitably at this time. However it's changing. In the 1970's it would have cost tens of Trillions. Now, fifty years later, it's down to merely tens of Billions. We can't really pull it all off yet safely. If attempted in the next ten years, there will be colonists dying. Maybe all of them. But work already planned on the Moon will make it get easier. Eventually, it will get easy enough that a colony can pay for itself over over a projected 50 year period. That's when it will happen.
Now? Probably not. But someday. Certainly before 2100.