Checking for EXTERNAL database and table existing by Drupal 7 API. First function check both, while second and third are just wrappers for initial one.

external_db_check.php

<?php

$db_connection_data = = [
    'host' => 'localhost',
    'database' => 'example_db',
    'username' => 'root',
    'password' => '',
    'driver' => 'mysql',
  ];

//This function checks DATABASE and\or TABLE existing.
//The main idea how to check if DATABASE exists (by Drupal DB API, not pure PHP) is to check any table existing (by db_table_exists()),
//and see what happened. Function db_table_exists() returns TRUE or FALSE if DATABASE accessible, or throw an exception if
//there are no DATABASE, or it is not accessible.
//$db_connection_data is the same config array like in "settings.local.php", $table_name can be any string if we check only for DATABASE existing.
function database_and_table_existing_check(array $db_connection_data, $table_name) {
  $response = [];

  if (!isset($db_connection_data['host']) || !isset($db_connection_data['database']) || !isset($db_connection_data['username']) ||
    !isset($db_connection_data['password']) || !isset($db_connection_data['driver'])) {
    $response['error'] = 'Incomplete settings array. Check settings array (variable $db_connection_data in database_and_table_existing_check() function).';
    return $response;
  }

  Database::addConnectionInfo('temporary_db_key', 'default', $db_connection_data);

  //Set active another\external database:
  db_set_active('temporary_db_key');

  try {
    //Function db_table_exists() returns TRUE if table exists, FALSE if not,
    //and throw an exception if connection error or DB not available - that is why we do this through try-catch.
    //@ before db_table_exists() is because this function not only throw an exception, but also generates warning.
    $is_table_exists = @db_table_exists($table_name);
    if ($is_table_exists) {
      $response['table_exists'] = TRUE;
      $response['db_exists'] = TRUE;
    } else {
      $response['table_exists'] = FALSE;
      $response['db_exists'] = TRUE;
    }
  } catch (PDOException $e) {
    $response['db_exists'] = FALSE;
    $response['table_exists'] = FALSE;
  }

  //It's very important to revert active database to default!
  db_set_active();

  return $response;
}

//This is just a wrapper for database_and_table_existing_check() for checking if DB exists.
//Returns TRUE or FALSE.
function is_db_exists(array $db_connection_data) {
  $response = database_and_table_existing_check($db_connection_data, 'dummy_table_name');
  if (isset($response['db_exists']) && $response['db_exists'] === TRUE) {
    return TRUE;
  } else {
    return FALSE;
  }
}

//This is just a wrapper for database_and_table_existing_check() for checking if TABLE exists.
//Returns TRUE or FALSE.
function is_table_exists(array $db_connection_data, $table_name) {
  $response = database_and_table_existing_check($db_connection_data, $table_name);
  if (isset($response['table_exists']) && $response['table_exists'] === TRUE) {
    return TRUE;
  } else {
    return FALSE;
  }
}