Is it possible for a differential in a Cech complex to be an isomorphism? If so, would that imply that the homology at the object right before the differential is trivial?
I have a very weird open covering of that results in many objects in the Cech complex being isomorphic, so I am wondering if there is some error.
A chain complex and its homologies is a general algebraic construction. Once you have an isomorphism $C_i\stackrel{\partial_i}{\to} C_{i-1}$, then the $i$th homology $H_i$, being the kernel of $\partial_i$ divided by the image of $\partial_{i+1}$, is trivial (as the kernel is trivial).
By the way, the Čech complex associated to the empty covering of the empty set consists of a chain of trivial groups and all $\partial_i$ are isomorphisms. Surely, you can find many nontrivial cases as well.
Unfortunately, I don't understand the last sentence of the question.