Depending on the exact properties of magic in your world, there may be various methods of performing fire magic...
Basing the magic mostly on real life physics, you'd need three items to create fire:
- Heat or energy
- An oxidiser
I'll try to describe what would be theoretically nescessary in the context of magic - so i'll put it into an example of making a basic fireball or flamethrower-like spell.
Though i'll also put in the limit that the wizard won't have to carry extra ingredients with him in order to create said spell - that is, he can only use the components of the same air he breathes.
So, let's start with a nitrogen reaction as OP describes...
The nitrogen in the air reacts with oxygen to form nitric oxide:
and nitrogen dioxide:
Provided that magic only creates the heat, you'd need a lot of energy to make this happen - likely beyond whatever energy output a normal living being could ever provide. Especially if the magic does not provide a way to contain the reaction with some manner of telekinesis
First off, you face the main problem of having to split an atmospheric dinitrogen molecule into individual nitrogen atoms.
Dinitrogen molecules are, however, bound together with a very strong triple molecular bond.
One of the few forces of nature that is able to break that bond is electricity, and a lot of it - in the form of lightning.
So, in this context, along with creating a telekinetic enclosed space, the wizard could use the nitrogen as fuel, a lightning spell as energy source, and oxygen as the oxidiser.
Then again, using nitrogen in the reaction wouldn't likely produce a fireball, even with extra oxygen added to the telekinetic "container". If that were the case, even a small fire could burn off the entire atmosphere.
On the other hand, if the mage has molecular reconstitution in his portfolio of magical abilities - he could easily create something like dicyanoacetylene (C4N2), which can burn at almost 5000 degrees C with a bright, white-bluish flame...
Provided that magic could also provide a weak electrical field, the wizard could utilize water vapor that is suspended in the air. Using the electrical property of magic, along with aforementioned telekinetic "container", the wizard could concievably create and regulate a bubble of hydrogen-oxygen mix (as well as various dust particles to make the fire visible) and ignite it with wither a spark (~20 microjoules) or sufficient heat (around 570 degrees C).
Not only is hydrogen combustible, the spell could also be regulated to have an explosive effect. It can also react with the atmospheric nitrogen to produce NOx gasses, as well as nitrogen-hydrogen compounds like ammonia
So, to answer the questions:
What does this tell us us about fire magic with flames?
I'd say, that like in alot of electrical devices, you'd have to "cheat"; using a nitrogen reaction alone, would be prohibitively expensive on its own.
Our mage would have to use other means to create the desired reaction.
Also, if magic can only provide heat in and upon itself, simply warming a spot of air would be unlikely to create the reaction, unless our mage is able to contain and heat the air to plasma.
However, a common property of magic is that it's considered a form of energy. Provided that our mage can conjure at least some degree of electrical current, this already provides a good shortcut.
You do forget one thing with standard, day-to-day atmospheric air: You have the oxidiser, our mage has heat and energy - but the main constituent of the atmosphere - nitrogen (or rather dinitrogen molecules) - is not a suitable fuel source due to its normally inert nature.
Leveraging hydrogen, gained from electrolysing water vapor, would also carry the limitation of being nearly or completely unusable in dry environments unless the wizard were to carry a flask of water (steam -> electrolyse -> boom)
The key part is to raise the concetration of the desired element(s) in the telekinetic reaction bubble and keeping the reaction fed with fuel.
How hot is it?
Hydrogen can burn at up to 2,800 degrees C, provided the mage is skilled enough or able to provide enough hydrogen to create a stochiometric reaction with oxygen - that is a 2:1 mix of hydrogen-oxygen
What color would the flames have?
Pure hydrogen flames are a very deep blue and emit mostly ultraviolet light, that can just barely be seen in daylight - so a "pure" fireball would likely have the same bluish tinge.
The end colour can be regulated by our mage, however, as he has the freedom to include any kind of dust or particles in the reaction. This dust would likely create yellow-orange streaks in the fireball depending on the concentration.
Our mage could also concievably regulate the temperature of the reaction - which in turn can regulate the colour.
Is the reaction endo- or exothermic?
Most, if not all burn/flame reactions are exothermic