Am I right in saying this: If there is an unpaired electron, then there is no neutral charge in that atom and a negative charge is given since the electron doesn't cancel out, since there is no pair. So does that negative atom carry that unpaired negative charge? And does that atom/element, like cobalt attract things because of the negative unpaired electron and how it attracts a proton from a different atom? Someone please make it clear to me.
Your statement is unfortunately wrong, as the electrons all carry a negative charge (that you could say is balanced by the positive charge of the protons in the atom nucleus). Single atoms of an element are neutral but can have unpaired electrons.
Atoms with paired charges can be negatively charged, most common ions like Cl- have no unpaired electrons and the addition of the electron that formed the final pair added the negative charge.
Electron pairing is about spin, electrons with opposite spins pair together to form a much more stable arrangement releasing energy.