Dear Jana,
let me try to answer your questions:
1) No there are no restrictions on how much imbalance there can be. In fact, it's one of
the advantages of the mixed model approach!
However, cells with 0 cases will not be used in the model fit. The mixed model will
"automatically" take the varying number of cases into account in estimates and standard
errors.
2) The imbalance in several random factors will indirectly influence the standard errors
of the fixed effects estimates (as the fixed effects estimates are obtained through a
weighted least squares approach with weights reflecting the random effects structure in
the model).
As the structure of your data is hierarchical with fields, farms, and regions it makes
sense to me to have random effects
(1|region) and (1|region:farm)
in the model. Actually I don't see the point in testing a feature of the data that was
imposed by the design of the study or experiment. Just leave both terms in the models and
proceed to evaluate the fixed effects.
Random effects reflect structure in the data that is imposed by how the data were
collected or the underlying experiment designed and therefore it's rarely relevant to test
this part of the model (it's simply the framework within which we examine some interesting
explanatory variable). I guess, however, that other statisticians might a different
opinion about this issue.
I hope this explanation is useful?!
Christian