Can we use 'input geometry' on the REST Query of a map service, to search inside a polygon, circle, or rings?
I tried to use the Query (Map Service\Layer) guide line, but I didn't get any results.
Yes you can query by Polygon, just provide geometryType "esriGeometryPolygon" parameter. See nice sample here:
Query a feature layer | ArcGIS for Developers
Retrieving data ...