Delta packages are different from normal packages in only one way: instead of including all project items, they include only items that were changed after a certain date. Whether or not an item has changed is determined by taking the value of every item's
__Updated field and comparing it to the date/time you set in TDS project settings.
When generating packages, TDS also includes deployment property 'metadata' from ALL items in the project (whether they're included by Delta Packages or not). That metadata tells the update package installer, how to add/remove/update any set of items.
If an item was removed at some point, then the package won't contain it, however the package may contain it's parent, and the ChildSync deployment property metadata for that.
Therefore, if your project had the parent item, with child sync on, and you deleted that child item from your project, it will be deleted with the install of the package.
For Delta Packages, this remains the same, even without the parent being part of the delta package (i.e even if itself wasn't updated itself since the delta date that's been set).
So to the title of this question, YES, Delta packages can delete items, the exact same way complete packages generated by TDS can delete items.
To directly answer the bulk of your question: no, without being able to set the child sync options, update packages (delta or complete) can not delete items.The recursive deploy action along with ChildSync is needed for that.