The construction using "like" is an idiomatic construction in English, that's why it is hard to translate. There is no idiomatic equivalent in German which fits perfectly.
But from a grammatical point of view there is a very similar construction with "so" in German. This construction can be used for introducing a quote, either from direct speech or from thoughts.
It is important to mention, that the level of style of this phrase in German is probably lower than the "like"-phrase in English. For me it is hard to compare the style levels, though, because my knowledge of English is not deep enough.
From my limited knowledge of the English language, I believe that the English phrase would not be considered problematic concerning style. When I learned the "like" construction, it appeared to my ear a as a very informal construction, but I found that its use is rather common in English. But, I am not competent to give a proper estimation on the style level of this construction in english language.
The german construction with "so" would definitely be considered sociolect, I'd say. It is used especially in youth language. It's used also among adults, but it is very low style.
Anyway, be it as it may - the grammatical construction comes closest, that's why I mention this translation.
Before looking at the translations, one more comment seems necessary: I think, the third sentence is actually a different construction than the first and the second: In the third sentence the be like-construction is not used to express an utterance. You can recognise that the third sentence has a different construction, by omitting the "like": in the first and second sentence, it would leave us with an incomplete sentence, but in the third sentence it would just be "they were surprised". That's why the translation of the third sentence with the "so"-construction does not sound idiomatic in German, because it is only to introduce a quote.
The translation of your sentences using "so" would be like:
At first I didn't understand, but then I was like "wait, really?"
Zuerst hab ich es nicht kapiert, aber dann so: "echt jetzt?"
Everytime I ask you to do something, you go like "no, please, I'm tired"
Jedes Mal, wenn ich dich bitte, mal was zu machen, du so: "Nein, bitte, ich bin müde".
When I did my coming out, people at the table were like, really surprised, you know.
*Als ich mich geoutet habe, waren die Leute am Tisch voll so: echt überrascht.
As I said above, the third translation is not idiomatic. The like has a different function here: It does not introduce a quote, but expresses vagueness. Hence a better translation into German would be to use an adjective which expresses uncertainty or vagueness, like irgendwie:
When I did my coming out, people at the table were like, really surprised, you know
Als ich mich geoutet habe, waren die Leute am Tisch wirklich irgendwie überrascht.
Take note that the use of "so" is able to replace the verb of saying or thinking. This is similar to the English construction with "like", where the verb of utterance is also not very important and can be replaced by to be or to go.
The close grammatical parallels and the fact, that the "so"-construction is coming from youth language makes me wonder, if it is an anglicism in German.