They can indeed. This site talks about [different types of essays][1], all of which I have seen taught in schools.

These four types are expository, persuasive, analytical, or argumentative.

The latter two are of the type commonly considered to be typical of academic essays.

The **analytical essay** should, as the name suggests, present a formal analysis. One should:

> Be “objective” [and] avoid using the first person too much

and one should not:

> Use slang or colloquial language (the language of informal speech)
> [or] contractions.

An **argumentative essay** should

> prove that your opinion, theory or hypothesis about an issue is
> correct [and] be a topic that you can support with solid evidence

(and, by extrapolation, use formal language as for an analytical essay).

An **expository essay**, however, is more about personal response: 

> Your reaction to a work of literature could be in the form of an
> expository essay, for example if you decide to simply explain your
> personal response to a work. The expository essay can also be used to
> give a personal response to a world event, political debate, football
> game, work of art and so on

and, in a **persuasive essay**:

> You are presenting an opinion and trying to persuade readers 

but in a less scientific way than in an argumentative essay.

Generally, there are still conventions that separate a personal essay from [creative nonfiction][2]; the latter might rely on narrative or characterizing devices, whereas the essay is likely to be more formal and use more straightforward, less "creative" devices. 

Editorial opinion pieces in a newspaper can be classed as essays. Typically, they can use slang and first person, but don't **tend** to play with nonstandard syntax or narrative devices too much. Of course, there is a lot of leeway that depends on the newspaper's style and intended audience, among other things. 

[The American Scholar][3] has a collection of essays that show how personal and nonacademic an essay can be.

As for whether an essay can be fictional:

I think the idea is that an essay will be based on truth, but can be subjective, while fiction would be called a story/creative writing or similar. I can't find a good source to back this up at present, though.
  [1]: http://access-socialstudies.cappelendamm.no/c319365/artikkel/vis.html?tid=382115
  [2]: https://www.creativenonfiction.org/what-is-creative-nonfiction
  [3]: http://theamericanscholar.org/dept/essays/