I currently have both a Drupal and a WordPress site. Is there a module that can display the latest posts from my WordPress site, or will I need to write my own?
The Aggregator module that ships with Drupal core can pull in posts from the WordPress site's RSS feed. For more functionality and overall flexibility, you could use the Feeds module.
Yes, there is.
Assuming Wordpress makes your posts available using RSS, the module built-in within Drupal's core may be an answer: Aggregator Drupal module