Can a person happen? Is "Zodanga happened" correct?

Formally this can be analysed as an example of metonymy, a figure of speech in which a term that denotes one thing is used to refer to a related thing [Wikipedia].

Here, the name Zodanga (which normally refers to a person) is used to refer to his actions (which are a related thing).

— What happened here?
— Zodanga arrived and caused mayhem.

The “causing mayhem” is what happened; Zodanga is so closely related to that (as he was the cause) that his name is used to refer to the whole situation.

Similarly with Silvia in your example. It’s actually perfectly possible to use that construction if Silvia has a reputation for doing something particular. You refer to the person, and imply that what has happened is what they are famous for.

It is correct (as you note in the question) that it’s normally limited to bad events, and becomes slightly humorous when applied unexpectedly to someone’s good reputation:

— What happened here?
— J. Paul Getty happened. That’s why they’re all waving $50 bills.

Even when used of bad events, this particular construction—using a person’s name metonymically with the verb happen—is rather informal and should be used mainly in informal contexts. Other types of metonymy naturally have their own level of formality, with some being very frequent in more formal or technical contexts; for example:

HSBC dropped nearly ten points yesterday

– is something you’re quite likely to hear in a business meeting discussing stock prices. It is metonymy because HSBC here refers not to the bank or the corporation itself, but to the price of HSBC shares.

In addition to Andrew's very fine answer, I'd like to point out that 'Person's name happened' can also signify the start of a relationship or meeting someone who made an impression.

I was just living my life, and then, all of a sudden, Silvia happened.

It's slightly humorous and playing on the ungrammatical-ness. Something like, "Zodanga is such a larger-than-life person, that when I say 'Zodanga happened,' you understand he is a force and know exactly what I mean." Perhaps most analogous to "a tornado happened," as a tornado is a noun whose mere presence indicates event-like qualities.

I would also like to point out that Zodanga was, in fact, the name of a city in the story (John Carter is based on A Princess of Mars by Edgar Rice Burroughs). This doesn't invalidate the accepted answer, except perhaps the example.

— Zodanga happened.

— Zodanga arrived and caused mayhem.

— Zodanga's military arrived and caused mayhem.

The second version, which is given in the accepted answer, is actually still acceptable, as long as you realize that Zodanga is still a metonym for "Zodanga's military" or perhaps "Sab Than," the leader of Zodanga and the name of the primary antagonist of the movie.

An analogous situation is often heard in U.S. political news, where you might hear "the White House" in reference to the current President or his administration, "Capitol Hill" in reference to U.S. Congress, or even "Washington," which actually means Washington, D.C. and refers to the U.S. Government in general.

Remember the once-running Yahoo chat rooms? Now:

A: What happened to Yahoo chat rooms?

B: Facebook happened!

The same story here.