Quarter hour calculation in time dimension

Answer to question 1 in the query below.

I don't understand question 2, especially the part "then the next 15 minute chunk will be 12:47 AM".

But maybe you can use the answer to question 1 for question 2 as well.

Your original with added step QuarterTime, also added to step setDataType. 

let
    Source = List.Times(#time(0,0,0) , 1440, #duration(0,0,1,0)),
    convertToTable = Table.FromList(Source, Splitter.SplitByNothing(), {"DayTime"}, null, ExtraValues.Error),
    createTimeKey = Table.AddColumn(convertToTable, "TimeKey", each Time.ToText([DayTime], "HHmmss")),
    hourIndex = Table.AddColumn(createTimeKey, "HourIndex", each Time.Hour([DayTime])),
    minuteIndex = Table.AddColumn(hourIndex, "MinuteIndex", each Time.Minute([DayTime])),
    QuarterTime = Table.AddColumn(minuteIndex, "QuarterTime", each #time(0,0,0)+#duration(0,0,15*Number.IntegerDivide(Duration.TotalMinutes([DayTime]-#time(0,0,0)),15),0)),
    setDataType = Table.TransformColumnTypes(QuarterTime,{{"DayTime", type time}, {"TimeKey", type text}, {"HourIndex", Int64.Type}, {"MinuteIndex", Int64.Type}, {"QuarterTime", type time}})
in
    setDataType