I'm the plugin author and I've updated the plugin as per your suggestion so that row toggling now happens by default only when the icons are clicked.

"I need to be able to differentiate between selecting the row itself and the tree controller, column name, etc. As well as bringing through critical row information (title, salary, and so on)."

To highlight the currently selected row and log the selected row info:

    const dt = $('#my-table').DataTable();

    $('#my-table tbody').on('click', 'tr', function () {

        const $row = $(this);
        if ($row.hasClass('selected') ) {
            $row.removeClass('selected');
        }
        else {
            dt.$('tr.selected').removeClass('selected');
            $row.addClass('selected');

            console.log(dt.row($row).data()); // data in selected row
        }

    });

Similarly to log info about a particular cell on a click event:

    const dt = $('#my-table').DataTable();

    $('#my-table tbody').on('click', 'td:not(.tt-details-control)', function () {
            console.log(dt.cell($(this)).data()); // data in selected cell
        }
    });

Hope this helps!

Format your HTML properly with <tr>...</tr> around <th>..</th> (& end <th> properly as shown) & add <thead>..</thead> around head & <tbody>...</tbody> around body and add code to auto add missing <td></td>:

$(document).ready(function() {
    // Add missing <td></td> according to number of columns found in head
    var numTd = $('th').length;
    console.log('Number of columns found in head =: ', numTd);
    $('tbody tr').each(function()
    {
        var i = $(this).find('td').length;
        while(i < numTd)
        {
            $(this).append('<td></td>');
            i++;
        }
    });
    
    $('.tree').DataTable();
} );

<link rel="stylesheet" href="https://cdn.datatables.net/1.10.13/css/jquery.dataTables.min.css" />
<link rel="stylesheet" href="https://cdn.datatables.net/responsive/2.1.1/css/responsive.dataTables.min.css" />
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="https://cdn.datatables.net/1.10.13/js/jquery.dataTables.min.js"></script>
<script src="https://cdn.datatables.net/responsive/2.1.1/js/dataTables.responsive.min.js"></script>

<table class="tree display responsive no-wrap">
  <thead>
    <tr>
    <th>name</th>
    <th>class</th>
    <th>Remark</th>
    <th>xyz</th>
    <th>zyz</th>
    <th>zyz</th>
    </tr>
  </thead>
  <tbody>
    <tr>
    <td>somethigThing</td>
    <td>somethigvalue</td>
    </tr>
    <tr>
    <td>somethigThing</td>
    <td>somethigvalue</td>
    </tr>
    <tr>
    <td>somethigThing</td>
    <td>somethigThing</td>
    <td>somethigvalue</td>
   <td>somethigThing</td>
    <td>somethigThing</td>
    <td>somethigvalue</td>
    </tr>
    <tr>
    <td>somethigThing</td>
    <td>somethigvalue</td>
    </tr>
    <tr>
    <td>somethigThing</td>
    <td>somethigvalue</td>
    </tr>
    <tr>
    <td>somethigThing</td>
    <td>somethigThing</td>
    <td>somethigvalue</td>
    <td>somethigThing</td>
    <td>somethigThing</td>
    <td>somethigvalue</td>
    </tr>
  </tbody>
</table>