Will it be legitimate to say something like "We did nothing much this evening. John drank beer. Bob and I played cards. Et cetera."
Rules are social prescriptions, not products of physical law. And they're often made up out of "whole cloth"/thin air by people with not much imagination but a very real desire for social power. The trick is to obey the rules if you'd rather not pay any social cost there might be for not doing so. But make your own choices otherwise.
The short answer is: yes, you can make it into its own sentence. Done that way it's considered informal, but because it's perfectly understandable, it's legitimate. Just like the breaking of other similar "don't" rules.
Et cetera (commonly abbreviated to etc.) is used at the end of a list to denote that it can continue but you chose to leave it there, possibly because it would be too tedious to continue the list, or it would be boring for the person hearing or reading it. (See this webpage)
So therefore et cetera would not form a sentence or be at the beginning of the sentence and the correct way of writing your passage would be
We did nothing much this evening. John drank beer, Bob and I played cards, et cetera.
using commas to form the list, and the passage would more commonly be written in the form of
We did nothing much this evening. John drank beer, Bob and I played cards, etc.
While et cetera most commonly appears appended to a list and follows a comma, it can indeed form a sentence.
This occurs more often in transcriptions of speech and in genres such as fiction, but is not necessarily informal nor is its use restricted to speech.
From a work of fiction published in a recognized literary magazine (House is personified here):
House also fell in love with its neighbor. It knew love was usually built of commonality. The stoic romance of wartime construction made the most sense to others of similar build. Row houses had shared interests, like walls or French drains. Et cetera. House had other reasons.
Publication information May/Jun2015, Vol. 128 Issue 5/6, p49-54. 6p.
Title Today's Smarthouse in Love.
Author Pinsker, Sarah
Source FIC: Fantasy & Science Fiction
Somehow she thinks I think she's ashamed of me. My job. My mother was a waitress. Et cetera. "What's wrong with you?" she says.
Publication information Summer2013, Vol. 121 Issue 3, p331-342. 12p.
Title THE WRECKER.
Author FINNEY, ERNEST
Source FIC: Sewanee Review
Nice shoulders call for a boat neck. Gorgeous legs? "Play on that," she urges. Nice hands? Rings. Nice jaw? Earrings. Et cetera.
Date 2007 (20070422)
Publication information STYLE; Pg. F6
Title DESIGNING WOMAN; ; Moving from muse to main attraction;
Author Carolyne Zinko, Chronicle Staff Writer
Source NEWS: San Francisco Chronicle
But Forbes has become the indisputable second. He is the choice of 17%, a sizable step up from the 4% who picked him before. As for the rest of the G.O.P. field, every one of them is still an et cetera. Phil Gramm 9%, Pat Buchanan 6%, Lamar Alexander 3%. Et cetera. More than a third of Forbes' current supporters in the poll preferred Dole last November.
Date 1996 (19960212)
Title WHO IS SWITCHING TO FORBES AND WHY
Author RICHARD LACAYO
Source MAG: TIME
His attorney, for instance, will say he hasn't broken the law since he got out of prison except for jumping the parole. Hasn't changed his name here in Washington. Hasn't gotten a different Social Security number. And he's been a model citizen. Et cetera. But what gets lost here in situations like this, I think, is the families of the victims.
Date 1992 (19920220)
Publication information STYLE
Title Lives Of the Killer Preacher;
Author Paul Hendrickson, Washington Post Staff Writer
Source NEWS: Washington Post
What does that mean to you, Peggy? NOONAN# What it means to me is that he does not want to talk about ObamaCare. It is widely assumed that in 2014, the bad news of ObamaCare, the dislocations, the lost coverage, the price hikes, the premium hikes, etc. Etc. That all of this will continue.
Date 2014 (140105)
Title FACE THE NATION for January 5, 2013, CBS
Source SPOK: CBS
That is why theologians for centuries held the intent to procreate essential to the rectitude of sexual intercourse. That is why so many church Fathers considered intercourse during pregnancy almost sacrilegious. Etc. I now turn to my second point:
Date 1993 (Sep)
Publication information Vol. 169, Iss. 8; pg. 7, 5 pgs
Title Paul VI was right--Comment/reply
Author Flannery, Kevin
Source MAG: America
"I'm leaving Captain Clear-cut. After twenty-three years. Goodbye half-million-dollar house on the Rim. Farewell cook and maid. Etc."
Date 1996 (Winter)
Publication information # . Vol. 22, Iss. 4; pg. 34, 39 pgs
Author McIlroy, Christopher
Source FIC: Ploughshares
All examples extracted from search results on the Corpus of Contemporary American English at http://corpus.byu.edu/coca/ (on . et cetera . and on . etc . Accessed today.
Some people will argue that such cannot be sentences or "full sentences" by some definition--such represents a limited perspective. For a fairly comprehensive treatment on definitions of sentence, see https://wikipedia.org/wiki/Sentence_(linguistics).
In the example you give, the final "sentence" is actually a sentence fragment, punctuated as a sentence.
That does not make it wrong. Good writing can contain sentence fragments punctuated as sentences.
This particular example is written in a much more staccato style than you would get if you turned it into proper sentences. It may be intended to evoke the informal atmosphere of that evening with John, Bob, and the narrator. One may imagine that over the course of the evening they occasionally grunted a few words (often not complete sentences) at each other.
This sort of thing is very much a matter of writing style and having a good "ear" for how native speakers communicate. Even native speakers can find it difficult to write well in such a fashion. It is not clear that even this particular example is successful, although seeing it in the context of a larger work might make it appear less unnatural.
Bear in mind that et cetera means "and other things", and as @Chris says, it is normally placed at the end of a list.
If we play around with the phrase "and other things", we could put:
Bob and I played cards. And other things.
But it looks a little odd. Using an ellipsis would be better (and more suggestive):
Bob and I played cards... and other things.
But as for using etc cetera (or etc.), I would use it conventionally:
Bob and I played cards, chess, etc.
The phrase "Et cetera." written and punctuated as a sentence by itself is not standard grammar, but that doesn't mean that it's "wrong". I would not regard your prose as confusing, nor as suggesting a lack of education on the part of the writer. Putting each person's actions in a separate sentence suggests that the people were indifferent to each other's activities, but precludes putting "etc." in the same sentence as any of those activities. Writing "Et cetera" as a stand-alone sentence is a nice way of indicating that the "narrator" didn't happen to notice what all the people were doing because it wasn't important. Whether or not such usage is standard grammar, I like it here.
As a parting note, "etc." should almost always be abbreviated except when it is used as a stand-alone sentence, in which case it should be written out.