When should I use 丁寧語 (polite language) as opposed to casual language? What should be used on the internet without sounding like a jerk or the president? Examples would be greatly appreciated. ありがとう！

質問を翻訳

Usually we use 丁寧語 when we talk to people who met first. If you used casual language at this time, you might looks childish.
On the Internet, we often use casual language when we write as a diary, write as talking to ourselves or write a message to close people.