The scary temperatures you see in news headlines are basically unaffected by the fires. Wikipedia has a good overview:

https://en.m.wikipedia.org/wiki/Global_surface_temperature

The overall issue with global warming is not that one place gets super hot once and sets a record. Otherwise I could make news headlines by setting my house on fire and getting "hottest temperature ever! (at my house)". Those local hotspots of fire will affect the average global temp only a tiny bit, because the earth is a big place and there's lots of places not currently on fire. The thing to worry about is the reverse actually: because the earth is warming, fires are increasing everywhere, and then everybody will be next to a fire on that blessed record-setting day.

Approximately two

Well, the record high temperatures are what cause the forest fires so we do have to take that into account. And the radiant heat that the fire gives off dissipates with the inverse square law so that limits it's contribution. Really it seems that the only major contributing factor to the increased heat, other than the effects of the already high ambient temperature and thus the decreased apparent humidity, are the excitation of the air molecules as they are transformed from elemental oxygen and plant matter into hydrogen hydroxide and carbon dioxide, along with other molecules due to incomplete combustion and contaminates. Overall I think a safe bet would be 2.

As soon as the thermometer stops going down and levels out while moving away from the fire I guess.

I'm going to say out of sight or someone will question it. That's not actually a tough requirement, most places aren't on fire at any given moment.

Well, assuming fire temperature to be 800C and temperature drop off close to 1/r² - you need to be approximately 28 meters away from a point source to feel 1 degree Celsius increase in temperature. Forest fire is not a point source, it's a wall of fire, but give it or take - 100 m away it you shouldn't feel any heat coming from the first fire.

Are the fires out of the environment?