Going through your logs, the error you are getting is
System.ArgumentNullException: Key cannot be null. Parameter name: key
at System.Collections.SortedList.IndexOfKey(Object key)
at System.Collections.SortedList.ContainsKey(Object key)
at Microsoft.SharePoint.Publishing.WebControls.ContentByQueryToolPart.AppendListTypes(SortedList sortedListItems, SPListTemplateCollection listTypes)
at Microsoft.SharePoint.Publishing.WebControls.ContentByQueryToolPart.populateListTypeDropDown()
at Microsoft.SharePoint.Publishing.WebControls.ContentByQueryToolPart.populateCBQControls()
at Microsoft.SharePoint.Publishing.WebControls.ContentByQueryToolPart.OnPreRender(EventArgs e)
at System.Web.UI.Control.PreRenderRecursiveInternal()
at System.Web.UI.Control.PreRenderRecursiveInternal()
at System.Web.UI.Control.PreRenderRecursiveInternal()
at System.Web.UI.Control.PreRenderRecursiveInternal()
at System.Web.UI.Control.PreRenderRecursiveInternal()
at System.Web.UI.Control.PreRenderRecursiveInternal()
at System.Web.UI.Control.PreRenderRecursiveInternal()
at System.Web.UI.Page.ProcessRequestMain(Boolean includeStagesBeforeAsyncPoint, Boolean includeStagesAfterAsyncPoint)
Ensure that "Title" column for all items in that list are not null. This is the mostly likely cause for this issue.