A short answer, is that to estimate interaction energy (which says if same charges attract or repel), you use propagators. Propagators come from the expression of Lagrangians. Finally, the time derivative part for dynamical freedom degrees in the action must be positive, and this has a consequence on the sign of the Lagrangian.
Choose a metrics $(1,-1,-1,-1)$
For instance, for scalar field (spin-0), we have ($i=1,2,3$ representing the spatial coordinate) the : $$S = \int d^4x ~(\partial_0 \Phi\partial^0 \Phi+\partial_i \Phi\partial^i \Phi)$$
Here, the time derivative part of the action is positive (because $g_{00}=1$), so all is OK.
When we calculate energy interaction for particles wich interact via a spin-0 field, one finds that same charges attract each other.
Now, take a spin-1 Lagrangian (electromagnetism):
$$S \sim \int d^4x ~(\partial_\mu A_\nu - \partial_\nu A_\mu) (\partial^\mu A^\nu - \partial^\nu A^\mu)$$
The dynamical degrees of freedom are (some of) the spatial components $A_i$, so the time derivative of the dynamical degrees of freedom is :
$$S \sim \int d^4x ~\partial_0 A_i \partial^0 A^i$$
Now, there is a problem, because this is negative (because $g_{ii} = -1$), so to have the correct action, you must add a minus sign :
$$S \sim -\int d^4x ~ (\partial_\mu A_\nu - \partial_\nu A_\mu) (\partial^\mu A^\nu - \partial^\nu A^\mu)$$
This sign has a direct consequence on the propagators, and it has a direct consequence on interaction energy, which is calculated from propagators.
This explains while same charges interacting via a spin-0 (or spin-2) field attract, while same charges interacting via a spin-1 field repel.
See Zee (Quantum Field Theory in a nutshell), Chapter 1.5, for a complete discussion.
My main question is why don't they affect other photons (unless colliding), shouldn't there be an attraction or repulsion by the exchange force? Or is it because I would need a QFT treatment in that case?
Photons are quantum mechanical entities, so yes, it is a QFT case. Any interaction between two photons goes through exchange diagrams. Two photon interactions occur, with very low probability because the diagrams are box diagrams with at least four 1/137 couplings depressing the probability . For light frequencies this is a very small number . This probability grows with energy so even gamma gamma colliders are envisaged.
Best Answer
You want something that does not exist - intuitive picture of physical process within a theory which is a demonstration of how far can one go with mathematisation of experience and ignoring intuitive pictures.
To study quantum electrodynamics you have to concentrate on its computational algorithms and neglect intuitive pictures, to study intuitive pictures you have to neglect QED.
Both are a good thing to study, just do not expect it is easy to make them consistent.