It's complicated, but perhaps a good comparison is newspaper headlines in the West: headings and phrases etc are compact without ever being slangy.

For example, a random ad on the Yahoo Japan homepage proclaims 予約来場いただくと進呈QUOカード５０００円分, which means "get a 5000 yen gift card if you come visit", but uses literary words like 来場 (arrival/visit) and 進呈 (gift) that would rarely be used in spoken Japanese, and even the polite いただく (to humbly receive) is in the short "dictionary" form instead of being conjugated out the way it would be in speech.

Depends on the type of websites. Most corporate websites are polite. In a message board like 2-chan, (unsurprisingly) not at all. Actually, some of the most vulgar Japanese that most people would never speak in real life can be only found on the Internet.

> I am completely ignorant here--how common is the polite form on websites?

Not op, but I can answer.

Not very common (reported speech, simulating speech, etc.).