How does my body temperature stay lower than ambient air temperature if it goes over 37°C? I know it's because of sweat, but why should the sweat over my skin get heat from the body, cooling it under the ambient temperature, instead of getting it from the hotter air all around me? 
If I think about it I would suppose that the sweat cools me down to ambient temperature, not lower. 
 A: 
If I think about it I would suppose that the sweat cools me down to ambient temperature, not lower.

This is true if the ambient air is at equilibrium with your sweat as regards the evaporation of the sweat, i.e. if it is saturated and the partial pressure of water vapour in the ambient air matches the natural vapour pressure at the equilibrium temperature.
Generally speaking, this is not the case ─ ambient air essentially acts as a reservoir of dryness (rather than a thermal reservoir at a lower temperature). Simply put, in any situation where the water vapour that evaporates from your sweat is insufficient to meaningfully raise the humidity of the air around you, the ambient air acts as a reservoir that will suck away that humidity.
The reason this maters is because evaporation takes energy, in the form of the latent heat of evaporation. So long as the air around you is dry, molecules can evaporate, and in this process they remove thermal energy from your body.
This is the reason why humid heat feels that much more opressive than dry heat: sweat cannot evaporate, because the air is already saturated by water vapour (or rather, for every molecule of water that evaporates and takes away its latent heat of evaporation, there's another one which comes in from the ambient air, hits the sweat, and joins in, depositing its kinetic energy as an equivalent latent heat of condensation).
