Also, it's not an issue with jQuery, but with some other FE frameworks like angularjs or vuejs you may/will run into an issue of Joomla been not aware of your input because it fed as real JSON, not a form. To get over it you'd like to add something like

$input = Factory::getApplication()->input;
$jsonInput = json_decode(file_get_contents('php://input'));
if($jsonInput){
    foreach ($jsonInput as $k => $v){
        $input->def($k,$v);
    }
}

Edit after question update

I've updated and commented two of the files where you're having problems

<?php
class modFileFinderHelper
{
    function __construct() {
        $this->currentdirectory = basename(JPATH_BASE);
    }

    // static methods only (the constructor above wont run for your ajax request)
    public static function FileFinderAjax() {
        // Here we get the data passed via ajax - note the values themselves have not been filtered!
        $data = JFactory::getApplication()->input->post->get('data',[],'array');

        // Presumably you would here do something worthwhile
        $response = [
            'something' => 12345,
            'your_path' => $data['path'],
        ];
        // Usually you can dump json and die, but there's a inbuild responder that'll encode things and die on your behalf
        // you'll see client side it also adds a few generic values like "success"
        echo new JResponseJson($response);
    }
}


jQuery(function($) {

    // grab a dom selector so we don't have to grab it each time we update
    var filefinder_container = $('#filefinder_container');

    function loadContent(path) {
        console.log({path: path});
        var request = {
            option       : 'com_ajax',
            module       : 'filefinder',  // to target: mod_filefinder
            method       : 'FileFinder',  // to target: function FileFinderAjax in class modFileFinderHelper
            format       : 'raw',
            data         : {path: path}
        };
        console.log(request);
        $.ajax({
            method: 'POST',
            data: request
        })
        .success(function(response){
            console.log('response: '+response);
            // we've got json but we need an object so let's parse it!
            response = jQuery.parseJSON(response);
            // and now we can use it like this.
            filefinder_container.text(response.data.something + " and " + response.data.your_path);
        });
    }
    console.log('onload, calling ajax with default path value: '+$('#filefinder_container').html());
    loadContent($('#filefinder_container').html());

    $('#path').on('change', function() {
        console.log('change event triggered, calling ajax with path value: '+$(this).val());
        loadContent($(this).val());
    });
});