The best gas for increasing temperature is 'safe greenhouse gas', which is rather fantasy but considering the setting maybe it makes sense.
Increased atmospheric pressure also contributes a little to temperature so that nitrogen is helpful, also if you have too much oxygen it'll be fatal so you need the nitrogen to dilute the oxygen pressure anyway.
You need a minimum .1 atmospheres of oxygen and it must be no more than 30% of the total gas.
So at minimum you need .1 atm oxygen and .234 atmospheres of any other gas to make a planet inhabitable. But the moon is very cold, with .1 atm oxygen and .234 nitrogen you only raise the surface temperature from -53 to -45 degrees.
If you use safe greenhouse gas instead of nitrogen you'll get a surface temperature of 10 degrees which will get you a colony cost of 0. I personally prefer to get the conditions more earth like and add heaps of nitrogen before the oxygen so I don't end up with people living in an unrealistic flammable environment like that.
In regards to your second question, all thats needed to start mining is to drop an automine onto a planet, comet or asteroid which has been geological surveyed and contains minerals. Minerals will build up in the bodies stockpile, you can either send a freighter there to pick it up or drop off a mass driver. Note that you can transport partial facilities , a freighter with a single small cargo hold will drop off a fifth of a mine or mass driver, or a standard cargo hold will deliver a whole building.
A mass driver can transport 5000 tons of minerals a year, which is 70 per 5 day increment, thats about 500 mines at the start unless theres multiple mineral types.
The body you're transporting to needs only 1 mass driver to catch any quantity of minerals. Once your mass driver is built youll see a dropdown in the mining tab to select which body to send it to, note you can set a reserve level which a mass driver won't dig into when sending out minerals.
Also note that eventually "civilian mining complexes" will start being set up in the solar system, they are a great cheap source of minerals as each CMC is worth 10 standard automines and they frequently expand to contain more.
CMC's will be built on bodies which at minimum contain sorium or duranium so I generally leave those alone untill a CMC has been setup on them then add my own mines. You usually see your first CMC's in under 10 years. My current game has 20 CMC's on 6 different bodies by year 12 giving me 50% more minerals at a cost of only a few mass drivers.
In relation to colonising the solar system, once you have researched nuclear thermal engines and cryogenic transports you'll eventually see civilian colony ships being built, shipping companies should have more than enough capacity to fill up your initial colonies in the solar system, also later when freighters are launched by civilians you can use them for transport of facilities, or just leave them alone to generate trade income. Note that one of the first items they will ship for trade is infrastructure, every colony creates it's own infrastructure and at the start you'll see earth creates more infrastructure for trade than you can afford with your factories, this is free expansion for your colonies if you just let the civilian freighters pick it up and ship it over.
Finally your question about using tankers; when you have fuel harvesters in orbit of gas giants they directly convert the sorium there into fuel so it can't be shipped by mass driver. Civilian shipping companies will also create harvesters too once you have the technology, collecting it is a matter of sending a ship to the civilian harvesters' task group and refuelling from it.
