Translation:She got offended because I forgot her name.
Why not was offended? Got offended sounds like the speaker is trying to imply that she shouldn't be offended. I don't think the Czech implies that.
No. The reference you gave describes the ambiguity of " She was offended" (see Nueby). It is one sentence that fits all, the passive and the state of mind.
Yes, but to become offended sounds so strange in English, unless you're talking about something that takes place for a length of time. Here it's one thing that happened at one time, it's not normal English. If there is no semantic difference between the two options in Czech why should we have to memorise these strange phrases in English? It can be really frustrating.