Can I fetch the tweet from Twitter if I know the tweet's id?

You don't need to use the API or know the user account.

Just drop it into the following URL format, where XXXXXXXXX is the tweet id.

https://twitter.com/i/web/status/XXXXXXXXX

Yes you can. Check out Twitter's API documentation

http://search.twitter.com/search.json?rpp=50?q=from:{twittername}

Just look at the public API.