The examples on the pictures are not the most natural laguage out there, and your confusion only means that you are starting to grasp the concept.
носить/нести are indeed the verbs to convey a general idea of carrying something. Нести is for one-way action, so it is normal to say "несёт" when someone BRINGS a thing somewhere or to someone (the act of carrying with "нести" has one destination, right?).

При- used with verbs of motion adds the meaning of "approaching", so приносить/принести is a more specific way to say "bring". However, it does look strange in "Собака приносит мужчине газету". All these sentences would be OK if they were used for habitual actions ("She brings him towels (every morning)", "Children bring their parents breakfast (sometimes)"). It doesn't look as a likely scenario in the case with the dog, that's why I percieve the sentence as slightly wierd.

You see, "приносить" is formed from general "носить". Even prefixed, it retains the meaning of "movement in several directions, or repeated action" or "motion in general". It doesn't express the idea of bringing something "now" and "to a certain place" (you have "нести" and "принести" for that purpose). It even took me.. long to understand what's wrong, but - yep, sounds strange for that very reason.