Hi Carma,
Could you check the service's REST endpoint and see if the numeric fields that are not available have 1. coded domains associated with them, and 2. whether one of the fields is a subtype field? Generally speaking, Dashboard filters out all non-numeric fields and fields with domains or subtype associated with them, the rest of them should be available to use.
For your second question, count is only for showing the number of all features based on your data source, and so it does not require the attribute fields. If you have a use case that needs to show feature count based on some attribute valuess, it would be great if you could share it with us.
Thanks and please let me know if you have more questions,
Tif