I am a graduate student in mathematics who has just begun his journey through Griffiths' Introduction to Electrodynamics (it is my first exposure to the subject). I am familiar with differential geometry (manifolds, (pseudo)forms, etc.) and I have some (vague; learning more in the meanwhile) acquaintance with Schwartz distributions. I would like to make sense of what the author is trying to achieve in the languages I already know, but I find even the first pages of the book not easy to follow.

I read about Coulomb's force $$ \vec F = q\,\frac{q^\prime}{\epsilon_0\, 4\pi\, \lVert x - x^\prime\rVert^2}\frac{x - x^\prime}{\lVert x - x^\prime\rVert} $$ exerted by a source (point) charge $ q^\prime $ located at some point $ x^\prime $ on a probe (point) charge $ q $ located at some other point $ x $ and about it's generalization $$ \vec F = \sum_{i = 1}^N q\,\frac{q_i^\prime}{\epsilon_0\, 4\pi\, \lVert x - x_i^\prime\rVert^2}\frac{x - x_i^\prime}{\lVert x - x_i^\prime\rVert} $$ to a bunch of source (point) charges $ q_1^\prime,\dots,q_N^\prime $ located respectively at $ x_1^\prime,\dots,x_N^\prime $. Afterwards I got introduced to the quantity $$ \vec E(x) = \sum_{i = 1}^N \frac{q_i^\prime}{\epsilon_0\, 4\pi\, \lVert x - x_i^\prime\rVert^2}\frac{x - x_i^\prime}{\lVert x - x_i^\prime\rVert}\label{discrete}\tag{$ * $} $$ called the electric field at the point $ x $ generated by the $ q_i^\prime $s. Everything's fine up to now. Please notice that I'm not following prof. Griffiths notations.

My problems arise when "charges distributed continuously over some region" pops out, and the expression for the electric field has to be derived in this case. The author lightheartedly claims that the sum $ \eqref{discrete} $ becomes the (formal?) integral $$ \vec E(x) = \int\frac{\mathrm dq}{\epsilon_0\, 4\pi\, \lVert \vec r\rVert^2}\frac{\vec r}{\lVert \vec r\rVert}\label{continuous}\tag{$**$} $$ where $ \vec r $ should be something like the displacement vector from the infinitesimal charge $ \mathrm dq $ (?) to $ x $.

Following the book, we learn that the $ \mathrm dq $ above can be "instantiated" in the following three cases:

1. if the charge is spread out along a line with charge-per-unit-length $ \lambda $;
2. if the charge is smeared out over a surface with charge-per-unit-area $ \sigma $;
3. if the charge fills a volume with charge-per-unit-volume $ \rho $.

Question 1. What mathematical object is best suited to describe the empirical idea behind $ \lambda $, $ \sigma $ and $ \rho $? I would say that $ \rho $ is a density defined on $ \mathbb R^3 $, and I would say with some confidence that $ \lambda $ and $ \sigma $ are also densities respectively on a $ 1 $-dimensional and a $ 2 $-dimensional submanifold of $ \mathbb R^3 $. Could all of them be thought as some sort of distributions? (I don't know the precise relation between the differential geometer's densities and the analyst's distributions, but I would like to learn more!)

Forgetting for a moment about the mathematical machinery I brought up above, and following again the book, we put

1. $ \mathrm dq = \lambda\,\mathrm dl^\prime $;
2. $ \mathrm dq = \sigma\,\mathrm da^\prime $;
3. $ \mathrm dq = \rho\,\mathrm d\tau^\prime $;

respectively, where $ \mathrm dl^\prime $ is the "element of length" (of our $ 1 $-dimensional submanifold?), $ \mathrm da^\prime $ is the "element of area" (of our $ 2 $-dimensional submanifold?) and $ \mathrm d\tau^\prime $ is the "element of volume" (of $ \mathbb R^3 $?).

If we stuck with $ \lambda $, $ \sigma $ and $ \rho $ being densities, we should probably think of the expressions $ \lambda\,\mathrm dl^\prime $ etc. in the following way: $ \lambda $ as a density can be written as $ \lambda = f\,\mathrm dl^\prime $, where $ f $ is a real valued function and $ \mathrm dl^\prime $ is the Riemannian density induced on the submanifold in question by it's pullback metric. Here we just call $ f $ again $ \lambda $.

Going on, $ \eqref{continuous} $ become $$ \vec E(x) = \underbrace{\int\frac{\lambda\,\mathrm dl^\prime}{\epsilon_0\, 4\pi\, \lVert {\color{red}{\vec r}}\rVert^2}\frac{\color{red}{\vec r}}{\lVert{\color{red}{\vec r}}\rVert}}_{\text{a strange line integral?}}\qquad \vec E(x) = \underbrace{\int\frac{\sigma\,\mathrm da^\prime}{\epsilon_0\, 4\pi\, \lVert {\color{red}{\vec r}}\rVert^2}\frac{\color{red}{\vec r}}{\lVert{\color{red}{\vec r}}\rVert}}_{\text{a strange surface integral?}}\qquad \vec E(x) = \underbrace{\int\frac{\rho\,\mathrm d\tau^\prime}{\epsilon_0\, 4\pi\, \lVert {\color{red}{\vec r}}\rVert^2}\frac{\color{red}{\vec r}}{\lVert{\color{red}{\vec r}}\rVert}}_{\text{a strange volume integral?}}\label{int}\tag{$***$} $$ respectively in the first, in the second and in the third scenario.

Question 2. What mathematical operation do the integrals in $ \eqref{int} $ denote? The equations above really don't make sense at all to me. I'm mostly bothered by two things:

1. if we were really (and sloppily) dealing with tensor densities, what we should get by integrating them are scalars. The integrals appearing above results in vector quantities and this instills me doubts about the nature of $ \lambda $, $ \sigma $ and $ \rho $.
2. assume we are integrating densities: it is not clear if the integrals in $ \eqref{int} $ are meant to denote the the "$ \int D $" (integral of a density $ D $) thing or the "$ \int f(x)\,\mathrm dx $" (Lebesgue integral of the function expressing the density $ D $ as $ D = f(x)\mu $ where $ f $ is a function and $ \mu $ is the Riemannian density induced by the metric) thing. I'll go for the second but I'm still confused.

Let's assume for the moment that I made sense of such an expression $ \vec E(x) = \int\text{stuff} $. After all of this there's another question I would like to ask.

Question 3. I have read that the electric and the magnetic fields are best rendered mathematically as respectively a (pseudo-)$ 1 $-form and a (pseudo-)$ 2 $-form. This has something to do with how the electric or the magnetic field components change when we make a change to our reference frame. Now, since we're basically working in $ \mathbb R^3 $ with the standard flat metric, it's obvious how to obtain a $ 1 $-form (or a $ 2 $-form, eventually) from a vector field $ \vec E $. But... this doesn't feel elegant enough to me. What if I wanted to deal with $ 1 $-forms directly? What would these computation look like in this case?

In other (maybe a little silly) words: what is the mathematical object that "integrated" (whatever that means) gives a $ 1 $-form? What is the mathematical object that "integrated" (whatever that means) gives the $ 1 $-form corresponding to $ \vec E $?

I would like to emphasize that this question is not about the physics, but about how to talk about the physics in modern mathematical terms. Fore reference, I'm roughly familiar with the material presented in Naber's two volumes book Topology, Geometry and Gauge Fields.