I think that the Cell Counter plugin does not support to select a ROI in an image and count only objects in that region. A workaround could be to define a ROI, then select an interesting image part and duplicate it. You can then run the cell counter on just this duplicated image part. Afterwards you can move the same ROI to another interesting image location and do the same there. This way you can select different locations all with the same size and get counts for all of them. But, this is indeed a really effortful and probably suboptimal workaround…
Concerning your second question, I am not sure if I got that right. But if you configure the Cell Counter plugin to count, let’s say 4 different object types, and one of these types does not appear in your image, i.e. gets no click, then in the results table there should appear a zero for that type. Isn’t that what you would like to have?