"Etcetera" becomes "etc.". It is an abbreviation.
ect., e.t.c., and e.c.t. are all wrong. The latter two are wrong because they represent acronyms of several words when "etcetera" is only one word, and the former is because "etcetera" doesn't start with "ect".
The confusion is due to some people incorrectly pronouncing the first "t" as a "c", "k" or "x" sound, which blends with the "c". This causes some confusion about the spelling of the word because using the "c" sound makes it sound like "ecetera", which makes people think that the abbreviation is "ect.", since that is the order of the letters in the self-inflicted misspelling of the word.
Doesn't help that most people can't spell the word "etcetera".