As small as it may be, does every 'thing' have a gravitational pull?
Newtonian: Not everything. But, only massive objects. Definitely not a photon (for example)...
GR: Every single energetic object curves spacetime the same way as massive ones, as a consequence of mass-energy $mc^2$. So, a photon does curve spacetime, as it contains energy. (as John mentioned...)
No matter how obsolete or negligible it may be, is it there? If so, how is it calculated?
For instance, let's take a proton. As it has charge, it is affected by electric & magnetic fields. So, Mass spectrometers can be used to deflect them. By applying a magnetic field perpendicular to the plane of the particle, it's made to traverse a circular path and in this way, it's mass can be determined.
What does 'it' affect?
This is the most important part. A massive body (though it's negligible) does affect another massive body. The force is much negligible. Even then, there's always the force. If there isn't any force like that, these interstellar clouds wouldn't have had any gravitational pull and no stars, planets, etc...
In Einstein's language, massive objects curve spacetime and even mass-less objects can exert a gravitational field, but Newtonian is good for this question..!