Delta-wye transformers are useful for a number of reasons. For transmission and distribution lines that are solidy-grounded, the delta-wye provides an excellent ground reference and source of ground current for clearing faults. Also, the delta-wye configuration isolates zero sequence current on each side of the transformer so that ground current relaying on one side does not see ground fault current for a ground fault on the other side.
Yes, there is a phase shift. For an ANSI transformer, the low side will always lag the high side by 30 degrees, regardless of the relative position of the delta and the wye windings.