How can I get newly created node of a certain type?
The equivalent of hook_nodeapi('insert') in Drupal 7 is hook_node_insert().  

function hook_node_insert($node) {
  db_insert('mytable')
    -&gt;fields(array(
    'nid' =&gt; $node-&gt;nid, 
    'extra' =&gt; $node-&gt;extra,
  ))
    -&gt;execute();
}