For topological spaces, a one object cover automatically satisfies the sheaf axiom because open inclusions are injective, which is equivalent to the projections of the kernel pair (= self intersection) being equal as arrows.
In the general context of sheaves on sites, however, it may happen that an arrow $U_i\rightarrow U$ in a coverage may not be a mono, in which case the projections from $U_i\times _U U_i$ will differ, and the sheaf condition will not be trivially satisfied.
Is there a geometric example of such a failure? Maybe something from the category of topological spaces with the étale topology? Can open embeddings of schemes fail to be monic?
An open embedding of schemes (called open immersion in algebraic geometry, alas!) is always monic, which answers your last question.
Of course a monomorphism of schemes needn't be an open immersion: a closed immersions is always a monomorphisms but extremely rarely an open immersion!.
(Actually the closed immersions are exactly the proper monomorphisms : EGA IV, 18.12.6.)