What are the composition and pressure of the exosphere?
Since this is the Earth Science stack exchange I assume you are talking about the Earth's exosphere, in which case the first issue to decide is how one defines the exosphere - there are several definitions. The lower boundary is straight forward - the thermosphere. However, the exosphere doesn't have a clearly defined upper boundary. You could argue that any gas molecule that is gravitationally bound to Earth is within the exosphere, but that depends upon the relative forces of solar wind and gravity, which is a variable. Some argue that the outer limit is defined by the geocorona at about 10,000 km, whereas the gravitational limit is more than 100,000 km, or arguably up to half the distance to the moon. 

The exosphere's composition varies from oxygen, carbon dioxide, hydrogen and helium at its base, to just hydrogen and helium in the outer limits.

The exosphere's pressure is about 0.0007 atmospheres at its base to virtually nothing in the outer reaches. In fact 'pressure' is a meaningless concept in the outer exosphere because individual atoms and molecules are kilometers to hundreds of kilometers apart, so they almost never interact - and pressure is really the intensity of molecular physical interaction.

A comparable argument applies to the exosphere's temperature. In the outer regions of the exosphere the same hydrogen molecule could have a daytime temperature of nearly 2000 Kelvin, and a night-time temperature close to zero Kelvin. In both cases the space would feel cold because there is next to nothing there to transmit heat.

The Earth's atmosphere is still evolving by processes dominated by the lithosphere / biosphere. Only hydrogen and helium are light enough to shoot up to the top of the atmosphere (think helium weather balloons). At the height and temperature of the thermosphere, or higher, they can both gain sufficient thermal kinetic energy from sunlight to gain escape velocity, which is why there is very little helium or free hydrogen in our atmosphere.