What is lost when kernels are defined in a category enriched over pointed sets? Traditionally kernels are defined in a category with a zero object, as equalizers with the zero maps. However, zero maps may be available whenever the category in question is enriched over pointed sets.
Is there anything to be careful about in this situation? Are there any facts/constructions involving kernels which really require a zero object? Does everything stay the same? What's the zero object really needed for?
Does this viewpoint "resolve" the "absence of kernels" in the category of unital rings, which has no zero object?
 A: Well, you don't need a zero object per se, and it makes perfect sense to define kernels for any category enriched over pointed sets.  However, assuming additionally that you have a zero object is a very mild assumption.
In particular, if $\mathcal{C}$ is a category enriched in pointed sets which has either a terminal object or an initial object, then it has a zero object.  Indeed, suppose $T$ is terminal (the initial case is dual) and let $A$ be any object.  There is a zero map $T\to A$, which I claim is the only map $T\to A$ so that $T$ is initial as well.  Indeed, if $f:T\to A$ is any map, its composition with the zero map $T\to T$ is the zero map $0:T\to A$.  But the identity is the only map $T\to T$, so the zero map $T\to T$ is the identity.  We thus have $f=f1_T=0$.
In fact, if $\mathcal{C}$ has all kernels and is nonempty, then it must also have a zero object!  Indeed, let $A$ be any object and let $Z$ be the kernel of the identity $1_A:A\to A$.  By definition, this means maps to $Z$ are in bijection with maps to $A$ whose composition with $1_A$ is zero.  That is, there is exactly one map to $Z$ from every object, corresponding to that object's zero map to $A$.  Thus $Z$ is terminal, and hence a zero object by the previous paragraph.
Furthermore, if $\mathcal{C}$ is a category enriched in pointed sets, we can very easily add a zero object to it: define a category $\mathcal{C}'$ by adding one new object $0$ to $\mathcal{C}$ with exactly one map to and from every object of $\mathcal{C}$, such that the composition of $A\to 0$ and $0\to B$ is the zero map $A\to B$.  Then $0$ is a zero object of $\mathcal{C}'$, and $\mathcal{C}$ is a full subcategory of $\mathcal{C}$ which is only missing this zero object.
As a result, there is almost no reason not to assume your category has a zero object when talking about kernels.  If you want all kernels to exist, then that automatically implies that a zero object exists unless your category is empty.  And if your category doesn't have a zero object, you can adjoin one to it without disturbing anything else you are likely to care about.
Finally, to address your last question, none of this is helpful for unital rings, since the category of unital rings is not enriched over pointed sets. (Indeed, if it were, then it would have a zero object, since it has a terminal object.)
