Ajax sends two empty arrays and one filled, PHP receives only the full one

You are using jQuery dataType options wrong!
The dataType: value should be 'json' not 'application/json' because your value will request with HTTP accept: */* but if you use 'json' it will be accept: application/json.

Option 1

Use content type application/json.

The contentType: should be 'application/json' or 'application/json;charset=utf-8'.

By this content type you will be now able to receive POST data in JSON but you cannot access them with $_POST because the data is not valid application/x-www-form-urlencoded.

Full code for client side:

var import_data = {
     action: 'start_import',
     sliced_batch: {
        rows_to_add: [],
        rows_to_delete: [],
        rows_to_update: [1,2,3,4,5]
     }
};

function ajax_call(import_data) {
    // ... processes

    jQuery.ajax({
        url: 'test.php', // url from php file
        type: 'POST',
        contentType: "application/json;charset=utf-8", // sending string with cyrillic (ukrainian lng)
        dataType: 'json', // want  to recieve json
        data: JSON.stringify(import_data), 
        success: function(response) { 
           // ... processes import_data ...
           
        },
        error: function(e) {
          // here is where I end up
        }

    });
}

Code for PHP:

$data = json_decode(file_get_contents('php://input'), true);
// use $data['sliced_batch'] to access `rows_to_add`, `rows_to_delete` etc.

Option 2

Use content type application/x-www-form-urlencoded.

With this content type, you will be able to access $_POST properly.

However, to use this request content type in header, the jQuery itself will be modify the value if it is empty jQuery will be just delete it!! So, you need to JSON string only sliced_batch property.

Here is the JS code:

var import_data = {
     action: 'start_import',
     sliced_batch: {
        rows_to_add: [],
        rows_to_delete: [],
        rows_to_update: [1,2,3,4,5]
     }
};

function ajax_call(import_data) {
    // ... processes

    // modify `sliced_batch` property to be JSON string to prevent jQuery remove empty properties.
    import_data.sliced_batch = JSON.stringify(import_data.sliced_batch);

    jQuery.ajax({
        url: 'test.php', // url from php file
        type: 'POST',
        // just remove contentType option. It is no need.
        //contentType: "application/json;charset=utf-8", // sending string with cyrillic (ukrainian lng)
        dataType: 'json', // want  to recieve json
        data: import_data, 
        success: function(response) { 
           // ... processes import_data ...
           
        },
        error: function(e) {
          // here is where I end up
        }

    });
}

PHP:

$sliced_batch = ($_POST['sliced_batch'] ?? '');
$sliced_batch = json_decode($sliced_batch, true);
// you can now access $sliced_batch['rows_to_add'], etc...

So, thanks once again to @vee for his explanation, but here's one more thing I'd like to share as it was crucial to get everything to work.
First, for json_decode method the JS object keys should be double-quoted, i.e. NOT

$bad_json = '{ bar: "baz" }';
json_decode($bad_json); // null

or

$bad_json = '{ 'bar': "baz" }';
json_decode($bad_json); // null

BUT

$bad_json = '{ "bar": "baz" }';
json_decode($bad_json); // array("bar" => "baz")

Second and most important! When dealing with WordPress it sets its own rules and shows focuses! Depending on what answer you'd like to get, you may want to use function wp_unslash(). Looking at the stringified data in console I saw somth like this:

"\u0421\u0435\u0440\u0432\u0435\u0440: \u0424\u0430\u0439\u043b\u0456\u0432 av_imp_p_WEB.csv \u0456 av_imp_p_WEB_previous.csv \u043d\u0435 \u0431\u0443\u043b\u043e \u0432\u0438\u044f\u0432\u043b\u0435\u043d\u043e. \u041f\u043e\u0447\u0438\u043d\u0430\u044e \u0456\u043c\u043f\u043e\u0440\u0442 \u0432\u0441\u044c\u043e\u0433\u043e 
// it is more common for contentType: "application/x-www-form-urlencoded"

It is the dirty work of WooCommerce (as I read from another's people opinion) and it hinders parsing it the right way, so my full code is: JS

 var import_data = {
          "action": "start_import",
           "sliced_batch": {
           "rows_to_add": my_data1,
           "rows_to_delete": my_data2,
           "rows_to_update": my_data3
            }
};

function ajax_call(import_data) {
 // ... processes
jQuery.ajax({ // ajax to php to save the file to uploads and move it to the plugin's folder
            url: start_import_ajax.url, // url from php file
            type: 'POST',
            //contentType: "application/json;charset=utf-8", //  what you send
            dataType: 'JSON', // what you would like as a response
            data: {
                "action": import_data.action,
                "sliced_batch": JSON.stringify(import_data.sliced_batch)
            },
                success: function(response) {
                    //response = JSON.parse(response); // if you'd like to console.log  what you've sent
                     console.log(response);
                 }
....

PHP

 $sliced_batch = wp_unslash($_POST['sliced_batch']);
    $sliced_batch = json_decode($sliced_batch, true);

    $result = start_import($sliced_batch);

    if($result == 0) {
        echo json_encode(["status" => 0]);
    } else echo json_encode(["status" => 1]);