In the first example, was able to is an alternative.
In the second example, could means that the speaker was in hearing distance of the phone, but implies that the speaker didn’t answer it, at least not immediately. If the speaker had said ‘I was able to hear the phone ringing’, the speech is more likely to continue with something like ‘. . . and so I went over and picked it up.’
In the third example, (1) is not necessarily wrong. It might occur in a sentence such as ‘The doctor said that after treatment he could return to work.’ (2) suggests that not only was he able to return to work, but that he did so.
Could and able to can be interchangeable, but the context will often decide which is chosen.
I don't speak German, but from the examples you've given I would say that the English and German usages are similar, but not identical. In English, it's almost always a rebuke, rarely a warning. It refers to forgetting about social rules, and your place within them. The expression can be mild or strong depending on the tone of voice, the speaker and the context.
In your examples:
John, we are going out to dinner tonight. Please don't forget yourself.
A mild rebuke. John has behaved badly in the past, and the speaker is gently requesting that he control himself tonight. Depending on the tone of voice, this could be a desperate plea from someone who feels slightly inferior to John (and therefore doesn't want to use stronger language) or a patronising order from someone who feels superior to John to the extent of treating him like a child.
I believe you forget yourself sir!
A strong rebuke from a social equal or superior. The speaker is very angry, and yet is trying to stay within the social rules himself.
I'm forgetting myself. I haven't offered you a drink yet!
The speaker has caught himself out in a social error, and is at once acknowledging his mistake and apologising for it. It's a mild self-rebuke, although tone of voice and context will show how embarrassed the speaker feels over the faux pas.
As you can see, it's all about context. The expression is almost never used in isolation, so the context is usually plain. The third one is probably the most common, though.
Finally, can this expression used as a threat or warning as in German?
It can, but it's not common. In English, such a threat would usually be wrapped up in a few more words. ("If you don't stop that, I might have to forget I'm a gentleman.") Like the second example, above, the speaker is trying to stay within the rules himself, while making it clear that he finds the other person's behaviour unacceptable. However, as social rules erode, this usage is becoming increasingly old-fashioned.
Best Answer
All four of the examples you have there are grammatical, and they are all acceptable in their specific contexts.
CAVEAT: This post is written on the fly, off the top of my head, etc., etc., and so, be forewarned.
There can often be a difference in meaning between the two expressions: "BE able to Verb" and "could". The context that they are in will often be a significant factor.
First, let's look at the four directly involved clauses in isolation:
Everyone was able to escape.
We were able to persuade them.
I (finally) could say something.
I (finally) could get my family back.
1/2: "BE able to Verb": The first two (#1, #2) could strongly imply that everyone did escape (#1), and we did persuade them (#2). If that type of strong implication is not immediately cancelled, or if the context had not already cancelled it, then that implication is then assumed to be a fact by the reader.
Though, the strength of that implicature could depend on the "Verb". For instance, for #2, it would be relatively hard to create a context where that implicature is not realized, that though we had the capability to persuade him, we chose not to. But for #1, that is not the case, as it would be easier in this case to come up with a context where that implicature is not fulfilled, e.g.:
Creating a corresponding context for #2 would be a bit trickier (if possible at all):
3/4: "could": As for your last two (#3, #4), it seems that there isn't any implicature (or else there is only a weak one) that "I" actually did say something (#3), or that "I" did get my family back. All those two are saying is that the possibility was there: for me to say something, for me to get my family back. There isn't anything there to strongly imply whether or not I did. The context they are in will probably have to explicitly say if I actually did.
Now, let's look at those four versions in their original contexts:
The context makes the very, very strong implicature that everyone did escape. If everyone didn't escape, then the writer has to immediately cancel that implicature.
The context makes the implicature too strong to be cancelled. That is, it is a fact that we did persuade them to come with us. But, it is still an implicature (a cancelable one) that they did end up coming with us--for it is possible that something then happened to prevent that.
It is the context--that I had earlier told her to get in line--that makes it a fact that not only could I say something but that I actually had said something. But a different context for that last sentence can easily show a situation where I didn't say something:
Here, the context has it pretty much open-ended: that is, the reader doesn't yet know if I actually did get my family back. Here's a continuation where I don't:
CAVEAT: I did this post off the top of my head. And now, maybe I'll go and do a little checking to see if I made any major goofs. . . .