Here are some leads for 1. and 2. (this is far from a full answer, but slightly too long for a comment). I only went in the "disadvantages of this approach" direction, too so that's another reason it's not a full answer. 

1. 

a- A mathematical advantage is that when you go to higher category theory/homotopy theory, then composition in the shape *does* matter (you need the naturality square for $f,g$ and $f\circ g$ to be compatible, and so on for higher morphisms). So in a sense what you describe is the consequence of the fact that in 1-category theory, equality of morphisms is a property and not extra structure. 
In particular, with the definition "shapes as categories", the higher generalization is clearer. This is to me the main disadvantage of that approach, especially in this day and age. 

b- Another advantage is that it allows for a unified framework (categories), where you don't have to introduce a new notion ("the category of diagrams with shape a directed multigraph") for something that already exists and is useful for many reasons (functor categories) - I guess this is somewhat the other side of the coin for your point 4. 

c- Related to that is Zhen  Lin's comment, that things indexed by categories somehow strictly generalize things indexed by directed multigraphs precisely because of your point 1 - this is also related to the previous point, that the category of diagrams is a special case of a functor category, but not the other way around (note that *co/limits* are not generalized that way, but categories of diagrams *are*)

d- Finally, a last advantage that I can see (related to your point about Kan extensions) is that many of the diagrams you naturally encounter - except for the "standard diagrams" : $\mathbb N$, co/products, co/equalizers, pullbacks/pushouts - are "naturally" indexed by categories. 
That is, often when you encounter a co/limit, it'll be because you had a functor in your hand, and wanted to understand something about it.

2. 

 a- Point 1.b- above seems to also be a pedagogical advantage as far as I can tell : you have strictly less things to remember/understand if diagrams are introduced as being indexed by categories. 

b- A certain number of arguments about limits and colimits involves taking composites $F(f)\circ F(g)$ where $cod(g) = dom(f)$, and are then just easier to state if you can reduce to $F(f\circ g)$. 

For instance, suppose you have an inverse limit indexed by $\mathbb N$ and want to replace it with the one indexed by $2\mathbb N$ for some cofinality argument - of course you can do the construction by hand, but it's easier to just define a functor $2\mathbb N\to \mathbb N$ than a directed multigraph morphism (which is a notion you have to introduce too anyways - this comes back to point 2.a- above). 

But cofinality arguments aren't even the only ones where you want to something like this (to give a second example, a condition you often meet is "for every $f$, $F(f)$ is blah" for some interesting notion "blah", which you would have to replace with "for every composable sequence, ...")