Is it possible to edit shapes while using a "definition query" or "query builder" in QGIS? I'm trying to do this in the same way I used to in ArcGIS.
It is not possible to edit a shapefile with a query. Following this ticket: http://hub.qgis.org/issues/2951, it is a matter of the underlaying OGR shapefile driver.
To solve the problem, store your data in a spatialite database, which can be queried and edited with no problems. That's what databases are designed for; a shapefile only comes with limited database-like functions.