air heated experiments
Yes!
as you blow out air using a small nozzle, the hot air cools down immediately.
I don't think that this is generally correct. There may be different opinions about the meaning of "small", if it's e.g. 10 mm or 1 mm diameter, or even less. But you can get hot air tools that achieve 500 °C even with really small nozzles.
A technical thermodynamics text book contains all necessary formulas and material properties to analyze your heater construction and would help you to design a working version.
Basic properties that have to be considered are heat capacity and heat conductivity of air. Also you should understand convective heat exchange in case of forced flow.
The first question is, if you actually achieve a sufficient air temperature at the outlet of your heater "coil". This should be checked with a thin wire thermocouple.
The second question is, if the hot air possibly cools down too fast, e.g. because the nozzle is kept cool.
As a third point, the hot air of course mixes with cold air, depends on where you are measuring the outlet temperature.
Finally, your temperature measurement method may be inappropriate, e.g. using a thick temperature probe.
But as most hot air blowers from a DIY market can achieve 500 °C, the problem seems solvable anyway.
P.S.: Considering the additional information, you gave in your second post:
You mainly have to know the exhaust air mass flow to calculate the required heating power. Most likely, you simply don't apply sufficient power.
Using an electric heater rather than a LPG burner can simplify the construction, but increases energy costs (I don't know your local prizes, of course).
Normally, thermal treatment of exhaust air in an industrial scale is required to use heat exchangers for energy recovery, if the energy costs shall be affordable. Unfortunately, the plant investment is respectively high.
As another comment, did you consider catalytic oxydation of the exhaust gas?