Category theory without objects is a little easier to formalise as a first-order theory. For instance, here is a first-order formulation of Lawvere's elementary theory of the category of sets, which adopts the strategy of identifying objects with their identity arrows.
That said, I'm not sure category theory without objects is any easier to think about.
Isomorphisms are absolutely essential to category theory, and in particular the idea that isomorphic objects are "the same" is perhaps the single most important concept in all of category theory. As you observe, you can define isomorphisms without requiring the existence of identities. However, this definition is still a bit problematic in a few ways. First, if "isomorphic" means "the same", you would expect it to be an equivalence relation, which it is not if identities don't exist. Second and more importantly, functors do not have to take isomorphisms to isomorphisms if you do not require that they preserve identities. This is a very serious problem for the intuition that functors should be some kind of "natural" operation: intuitively, an operation that can send isomorphic objects to non-isomorphic objects seems like it would have to be highly unnatural. I suspect this had at least something to do with Eilenberg and Mac Lane's decision to require that categories have identities (and that functors preserve them).
Here is another very basic thing that can go wrong in the absence of identities: you can have non-isomorphic terminal objects. Let $\mathcal{C}$ be a category with three objects $A$, $B$, and $C$ and exactly one map between any two objects, and let $\mathcal{D}$ be the non-unital category obtained by adjoining to $\mathcal{C}$ a new map $f:A\to C$ whose composition with any map of $\mathcal{C}$ is the unique map in $\mathcal{C}$ with the same domain and codomain. Then $A$ and $B$ are both terminal in $\mathcal{D}$, but they are not isomorphic (in fact, $A$ has no identity map).
Since any sort of universal property can be described as a terminal object in an appropriate category, this means that in non-unital categories, you cannot expect any universal property to define objects uniquely up to isomorphism. One way to fix this is to modify the definition of "terminal" to require that a terminal object have an identity map. Another way is to just require that all objects have identity maps, which holds in pretty much every example of interest.
Finally, on a more philosophical note, I would say that the existence of identity elements is actually a corollary of the "correct" definition of associativity. I would say that an "associative operation" is an operation which given any finite ordered set of elements $(a_1,\dots,a_n)$ gives you a "product" $a_1\dots a_n$ such that you can always drop parentheses (so for instance, $(ab)c=abc$, where the left-hand side denotes the binary product of (the binary product of $a$ and $b$) and $c$, and the right-hand side denotes the ternary product of $a$, $b$, and $c$). The identity element is then just the product of the empty ordered set. Considering "associative" operations which don't have identities is thus analogous to considering finite sets but not allowing the empty set. Of course, this is sometimes useful to do, but unless you have a good reason to, it is probably not the natural thing to do.
Best Answer
An arrow in the product category is a pair of arrows, one from each factor category (that is, $\text{Arr}(C\times D) = \text{Arr}(C)\times \text{Arr}(D)$). Since $\mathbb{2}$ has $3$ arrows, $\mathbb{2}\times\mathbb{2}$ has $3^2=9$ arrows. So yes, you found them all.