> What makes this cell "special" and how to make it an "ordinary" cell again?

It is the `CellGroupingRules -> {"GroupTogetherGrouping", 10000.}` option. Removing this option from input cell restores the default grouping in the case described in the OP. Generally whenever one uses the menu item "Group Cells", this option is set for all cells which should be grouped together. Apparently this is all what this option do.

> Is it possible to keep automatic grouping for generated cells if the input cell was manually grouped with some previous cell?

One simple way is to apply `"SectionGrouping"` by adding the option `CellGroupingRules->{"SectionGrouping",100}` to the `"Text"` cell. According to [this][1] answer, only `"SectionGrouping"` and `"TitleGrouping"` group `"InputGrouping"` (i.e. `"Input"` and `"Code"` cells). The disadvantage is that it will also group all further input cells what is undesirable. One non-perfect solution suggested in that answer is to add a delimiter having `CellGroupingRules->{"GroupTogetherGrouping", 100}` which will break up grouping for `{"SectionGrouping",100}` but not for `{"SectionGrouping",50}` (i.e. `"Subsubsection"` style). Here is how it can look:

[![screenshot][2]][2]

And here is a code which generates a template:

 NotebookPut@
 Notebook[{Cell[
 CellGroupData[{Cell["", "Text", CellGroupingRules -> {SectionGrouping, 100}], 
 Cell[BoxData[""], "Input"], 
 Cell["", "Text", Editable -> False, Selectable -> False, 
 CellFrame -> {{0, 0}, {0, 0.5`}}, ShowCellBracket -> False, 
 CellMargins -> {{0, 0}, {1, 1}}, CellElementSpacings -> {"CellMinHeight" -> 1}, 
 CellGroupingRules -> {"GroupTogetherGrouping", 100}, CellFrameMargins -> 0, 
 CellFrameColor -> RGBColor[0, 0, 1], CellSize -> {Inherited, 3}], 
 Cell[BoxData[""], "Input"]}]]}]

Another approach to the problem is demonstrated by [Carl Woll][3] in [this][4] answer. It has a drawback: output of `Echo` can't be auto-grouped with this method because it isn't supported by [`"GeneratedCellStyles"`][5].


 [1]: https://mathematica.stackexchange.com/a/152788/280
 [2]: https://i.sstatic.net/5rlZ9.png
 [3]: https://mathematica.stackexchange.com/users/45431/carl-woll
 [4]: https://mathematica.stackexchange.com/a/199441/280
 [5]: https://mathematica.stackexchange.com/a/71346/280