Trouble connecting stress and force in continuum mechanics with my concept of force from point mechanics

I'm not very familiar with continuum mechanics and have a hard time combining my knowledge of forces from simple mechanics with what I read about continuum mechanics.

Let's suppose we have a metal rod of a certain length and a quadratic cross-sectional area $A$ which is put under stress $\sigma$.

1) What exactly is the physical significance of the force $F = \sigma A$? My current understanding of forces is that they need a point of application - where would this point be? Is it a single point? All points of the cross-sectional area at once? The latter seems to conflict with the notion that force gets smaller if I consider only a part of the area:

If I partition the cross-sectional area $A$ into a number of smaller areas $A_i$, I can calculate forces $F_i = \sigma A_i$. Since $A = \sum_i A_i$, we also have $F = \sum_i F_i$. This makes it seem that $F$ is some kind of cumulative quantitiy and raises the question: what is the physical significance of the $F_i$?

Trying to generalize this further, we can also admit different values of stress $\sigma_i$ for the different areas $A_i$ and even make the areas infinitesimal so that we get a stress distribution $\sigma(x,y)$ (in order to simplify things let's suppose that we chose the distribution such that there's no net torque on the rod). What is the physical significance of the "force" $F = \int_A \sigma(x,y) dA$?

2) What's going on mathematically? Does the notion of force as a vector with a point of application need to be replaced by some kind of vector field in continuum mechanics (considering only stress in a single direction and ignoring additional complications related to the tensor nature of stress)? If yes, how can these "area forces" (I've also read the term "surface force") be combined with forces which have a point of application (for example with the weight of a point mass or a rigid body where the force can be described as acting on its center of mass)? In order to be combined, they need to be described by the same mathematical structure.

3) Does the notion of force somehow lose its meaning along the path which I sketched in 1) above? Is the integral quantity $F = \int_A \sigma(x,y) dA$ a force in one of the senses covered above or is it a quantity with units of force but without physical significance as a force?