Multiple inserts can be made using Fold, e.g.

    list = {1, 2, 3, 4, 5, 6, 7, 8, 9};

    Fold[Insert[#1, #2[[1]], #2[[2]]] &, list, {{"a", 2}, {"b", 4}, {"c", 6}}]

> {1, *a*, 2, *b*, 3, *c*, 4, 5, 6, 7, 8, 9}

Applied to the OP's data structure.

    shapes = Range[12];
    r = ReplacePart[Range[15], 2 -> "Nov 9 2017 13:24"];

    save = allEntries = {
        {r, r, r},
        {r, r},
        {r, r, r, r},
        {r},
        {r, r, r, r},
        {r, r, r},
        {r},
        {r, r, r, r},
        {r, r},
        {r, r, r},
        {r},
        {r, r, r, r}
        };

Demonstrating the OP's single insert function.

    allEntries = Table[Insert[allEntries[[j, i]],
        DateString[{allEntries[[j, i, 2]],
          {"Month", "Day", "Year", "Hour", "Minute"}}, "DayName"], 3],
       {j, 1, Length[shapes]}, {i, 1, Length[allEntries[[j]]]}];

    allEntries[[1, 1]]

> {1, "Nov 9 2017 13:24", "Thursday", 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

Resetting `allEntries` for the multiple insert method.

    allEntries = save;

    allEntries = Table[
       Fold[Insert[#1, First[#2], Last[#2]] &, 
        allEntries[[j, i]], {{DateString[{allEntries[[j, i, 2]],
            {"Month", "Day", "Year", "Hour", "Minute"}}, "DayName"], 3},
         {DateString[DatePlus[
            allEntries[[j, i, 2]], {1, "Day"}], "DayName"], 4},
         {DateString[DatePlus[
            allEntries[[j, i, 2]], {2, "Day"}], "DayName"], 5}}],
       {j, 1, Length[shapes]}, {i, 1, Length[allEntries[[j]]]}];

    allEntries[[1, 1]]

> {1, "Nov 9 2017 13:24", "Thursday", "Friday", "Saturday", 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

Much easier to use `ReplaceAll`.

    allEntries = save;

    allEntries = allEntries /. {a_Integer, date_, rest__} :> {a, date,
        DateString[{date,
          {"Month", "Day", "Year", "Hour", "Minute"}}, "DayName"],
        DateString[DatePlus[date, {1, "Day"}], "DayName"],
        DateString[DatePlus[date, {2, "Day"}], "DayName"], rest}

    allEntries[[1, 1]]

> {1, "Nov 9 2017 13:24", "Thursday", "Friday", "Saturday", 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}