dg.differential geometry - A comprehensive functor of points approach for manifolds

Here are two things that I think are relevant to the question.

First, I want to support Andrew's suggestion #5: synthetic differential geometry. This definitely constitutes a "yes" to your question

is there any sort of way to attack differential geometry with abstract nonsense?

--- assuming the usual interpretation of "abstract nonsense". It's also a "yes" to your question

Can we describe it as some subcategory of some nice grothendieck topos?

--- assuming that "it" is the category of manifolds and smooth maps. Indeed, you can make it a full subcategory.

Anders Kock has two nice books on synthetic differential geometry. There's also "A Primer of Infinitesimal Analysis" by John Bell, written for a much less sophisticated audience. And there's a brief chapter about it in Colin McLarty's book "Elementary Categories, Elementary Toposes", section 23.3 of which contains an outline of how to embed the category of manifolds into a Grothendieck topos.

Second, it's almost a categorical triviality that there is a full embedding of Mfd into the category Set${}^{U^{op}}$, where $U$ is the category of open subsets of Euclidean space and smooth embeddings between them.

The point is this: $U$ can be regarded as a subcategory of Mfd, and then every object of Mfd is a colimit of objects of $U$. This says, in casual language, that $U$ is a dense subcategory of Mfd. But by a standard result about density, this is equivalent to the statement that the canonical functor Mfd$to$Set${}^{U^{op}}$ is full and faithful. So, Mfd is equivalent to a full subcategory of Set${}^{U^{op}}$.

There's a more relaxed explanation of that in section 10.2 of my book Higher Operads, Higher Categories, though I'm sure the observation isn't original to me.