For the sentence 'I watched a movie after I had food', are the following translations equivalent? Additionally, is the 3rd translation incorrect (is the ...了...以后... usage valid)?
Briefly, I'd parse them as
以后 corresponds exactly to the English equivalent of 'after', while 了 has some more leeway in how it's used. All three of your sentences imply that you ate, and then you watched a movie (arguable for the first. The English equivalent would probably be "I ate food, then watched a movie"). I think 3 would be the closest in meaning to your original sentence. Although if you said that before a meal time, I'd assume you meant the upcoming meal, while if it was late at night, I'd assume that you'd already ate and watched.
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I'm not a native speaker, but I don't like the first sentence, I don't think it's correct.
To me the 2nd sentence implies a future event (I will watch a movie after I eat), whereas the 3rd sentence implies a past event (I watched a movie after I ate) although I would like to ask a native speaker whether there should be another 了 in that sentence such as 我吃了饭以后看电影了.
When translating sentences that are in the past into Chinese it is always best to use time markers such as "yesterday", "this morning," etc. It is time markers that put sentences in a "time context," not "了" or other particles. Strictly speaking your sentences are not equivalent, because they are ambiguous when it comes to tense. Sentence 2, for example, could be in the past, present or future tense depending on context. Something similar is going on in the other two sentences. As to how to translate your sentence, I would use the pattern I would use the pattern "先...再" (or 然后) and "了" or the word "接着" and "了."