As indicated in Jared's comment, one may find easy examples of 1), 2), and 4) by taking non-unital subrings (such as left ideals or two-sided ideals) of rings with identity.
The most interesting request here is 3). In fact, any finite nonzero associative ring $R$ (possibly without identity) without zero divisors is a field.
First, let's prove that $R$ in fact has an identity. Let $a \in R$ be a nonzero element. The function $f \colon R \to R$ defined by $\phi(x) = ax$ is injective, and since $R$ is finite, it's a bijection. Again, because $R$ is finite, this bijection must have finite order. Thus for some $d$, the function $\phi^d(x) = a^d x$ is the identity. It follows that $a^d \in R$ is an identity element.
At this point, it's a standard exercise to show that a finite ring with identity and no zero divisors is a division ring. (Hint: think about the function above for any $a \in R$.) And Wedderburn's little theorem states that any finite division ring is a field.