One can. The device to use is the 'zig-zag transformer'.

Yes adding a zig zag transformer to a delta transformation can provide an earthing reference

Here is an example

Zig-Zag Transformer and Transformer Vector Groups

Yes, In delta transformer, you can get a kind of neutral by center taping in one of the delta legs. This is done in USA for residential power supply. In residential power supply, each house is provided with two phases and mid point of these two phases that becomes as 220V/1110V power supply but not as 220V/127V in star connected phase to phase and phase to neutral.

Yes; you can create an artificial neutral by connecting a zig-zag transformer ( neutral former ,which is basically a star connected primary winding only ) to the delta transformation ungrounded power system for the purpose of monitoring the health of the insulation /earth fault in the system.

Alternatively,you can use a neutral earthing transformer ( basically a conventional star/delta transformer whose secondary winding is normally kept unloaded ) for the above purpose.