Does everything with mass or energy have a gravitational pull?

Yes, everything generates a gravitational field, whether it is massive or massless like a photon.

The source of the gravitational field is an object called the stress-energy tensor. This is normally written as a 4 x 4 symmetric matrix, and the top left entry is the energy density. Note that mass does not appear at all. We convert mass to energy by multiplying it by $c^2$ (as in Einstein's famous equation $E = mc^2$) and then put in the energy. So even a photon generates a gravitational field because although it has no mass it does have energy.

It's surprising what else is in the stress-energy tensor and can therefore generate a gravitational field. For example pressure and shear stress appear. It's even been suggested that a gravitational field could be generated by gravty itself i.e. the energy of the gravitational field generates the curvature that creates the field. The resulting object is called a geon, though I should emphasise that no-one has proved these could exist and most of us think they probably can't.

As small as it may be, does every 'thing' have a gravitational pull?

Newtonian: Not everything. But, only massive objects. Definitely not a photon...

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...