Radiation of heat is done through em waves which are massless particles. Being in direct contact with the air will transfer heat via conduction, or particles vibrating against each other - which is how the vast majority of heat loss will occur.

I’m not sure what you mean by in there but yes, the heat would be transferred to the environment.

E=m(c^2) describes how much energy is contained in matter. It’s useful for nuclear reactions, but your body isn’t a nuclear reactor and you aren’t consuming substantial quantities of radioactive isotopes, like uranium ore, that will decay on their own so it isn’t relevant here.

Producing heat isn’t where the mass goes though - mass is conserved. You only lose mass to energy in a nuclear reaction.