In a nested design (e.g., students in schools, ants in colonies, individuals in a species, leaves in a tree), ICC is defined as the proportion of variance among observations from the same subject (over total variation, i.e., variance within + residual variation). See, e.g., answer here
When in the design there are one or more categorical factors grouping several subjects (between-subjects factors, e.g., type of school, species of ants or trees), possibilities for heterogeneous variances appear. Residual variances can be modeled as heterogeneous in a mixed, random-intercepts model, for example as in (R code, nlme package):
lme(fixed = Y ~ A, random = ~1 | Subject, weights = varIdent(form = ~1 | A))
where there is then a single estimated variation among subjects, but as many residual variations estimated as levels in A.
(1) Is it valid/appropriate to estimate and report several ICCs, one for each level of the fixed factor? This would represent the correlation among observations from the same subject for subjects within each of the groups defined by the factor A.
(2) Moreover, if variances among subjects are modeled as heterogeneous among groups (this is, random intercepts for each level in A come from different normal distributions, see here), the answer to (1) is the same?