Multicategories vs Categories
Is there a way to recapture the additional understanding imparted by multicategories using higher categories?
Well I would say so. Multicategories are basically categories whose morphisms have multiple sources instead of only one. Bicategories generalize categories by adding 2-morphisms, i.e. relations among morphisms, this is completely different from adding multiple sources to the domains of the morphisms.
Is there a way to recapture the additional understanding imparted by multicategories using higher categories?
None that I'm aware of, but that should be expected since they provide different additions to categories (multiple sources vs morphisms among morphisms).
If not, then I would ask if a theory of higher multicategories exists and if the additional work of learning it over higher category theory is worth the understanding payoff.
Again not that I'm aware of.
For those familiar with it, does the theory of augmented virtual double categories have any significant ‘big picture understanding’ advantages over the theory of bicategories? What about compared to higher categories?
I'm not really familiar with that, but far I haven't seen lots of work on the subject. Probably time will tell.
Hope this helps.
Multicategories and bicategories, to me, are first of all completely orthogonal generalisations of monoidal categories, with virtual double categories as a common generalisation of multicategories and (strict) $2$-categories (they are to multicategories as categories are to monoids, or $2$-categories to monoidal categories). As for generalising simply categories, they are even more different, as multicategories are still a strictly associative structure while coherence issues appear for bicategories.
So, to your second highlighted question, I would say that the answer is no. A way to more precisely understand the difference, and a positive answer to your first question, lies in your final (bonus) question.
In addition to the aforementioned algebraic aspect, double categories have a large conceptual advantage over bicategories for formal category theory; in short, if you try to treat the objects of an arbitrary $2$-category as abstract categories, you will be lacking a lot of elements (the Yoneda structure coming from profunctors a.k.a. bimodules) to speak about limits in them, while double categories (at least the ones equipping their vertical category with proarrows) will give you enough. Virtual double categories are just the relevant generalisation for when bimodules do not compose, and the augmented version deals with the case lacking identity bimodules (e.g. for non-locally small categories). This is, in my opinion, the main conceptual payoff for (augmented) virtual double categories.
To finish, two technical generalisations, the latter of which was your second-and-a-half question:
Virtual double categories (though not the augmented ones, as far as I know), are an example of "generalised multicategories", something that can be defined relative to any monad acting on a virtual proarrow equipment. For this, see Leinster's book suggested in varkor's comment (chapters 4 and 5) or the more general and recent A unified framework for generalized multicategories by Cruttwell and Shulman.
There does exist a higher-categorical version of multicategories and even virtual double categories, defined (as "generalised non-symmetric $\infty$-operads") and put to great use (precisely to unify $\infty$-multicategories and double $\infty$-categories) in Gepner–Haugseng's Enriched $\infty$-categories via non-symmetric $\infty$-operads and follow-ups. A more general form, close to the generalised multicategories, is offered by Chu–Haugseng's formalism of algebraic patterns developed in Homotopy-coherent algebra via Segal conditions (see in particular section 9 with ex. 9.8). For the moment this is only used for algebraic aspects rather than formal (higher) category theory, but I would argue it is definitely worth learning (not over higher category theory, but as an extension of it).