More classes of rings spanned by units?

The study of rings (additively) generated by units is quite old, going back at least to the 1950's. The terminologies that you should look for include S-ring and k-good ring.

In case you haven't seen it, I can recommend Srivastava's A survey of rings generated by units as a good starting point to find out what is known. It will give you many good references, such as Raphael's Rings which are generated by their units and Henriksen's Two classes of rings generated by their units.

There are a lot of interesting things to say, especially along the lines of rings whose elements are sums of $n$ units. A $k$-good ring is one in which every element is a sum of $k$ units. Henriksen showned that every ring is Morita equivalent to a $3$-good ring.

For example, every linear transformation of a vector space over a division ring is a sum of two invertible transformations, excepting the case of a $1$ dimensional space over $F_2$.

From Srivastava's article:

Let X be a completely regular Hausdorff space. Then every element in the ring of real-valued continuous functions on X is the sum of two units [18]. Every element in a real or complex Banach algebra is the sum of two units [22].

Since rings of linear transformations have this nice proprety, it's natural to ask about von Neumann regular rings. It turns out, though, that it's possible for an element of a VNR ring to not be expressible as a sum of units. But not all is lost: all unit-regular VNR rings in which $2$ is a unit are S-rings (in fact, they are $2$-good!), and some self-injective VNR rings are S-rings.

I think that there is not much to say about a structural characterization of such rings, though. As you've noted, the condition does not pass meaningfully through important constructions. If polynomial rings aren't ever S-rings, and every ring is Morita equivalent to an S-ring, we have not learned much about structure.