"Elder" is softer in connotation than "older" - especially when referring to women. Even you sister does not want be referred to as an "older woman." "Elder" carries respect; "older" carries age. I don't agree that 'elder' necessarily conveys the idea of respect or softness. And we cannot refer to my sister as an 'elder' woman; 'elderly', yes, but that's a different word.
The "elders" of a tribe or society are the wise ones. Yes.
The "older ones" are the weak ones. Not necessarily.
It is not a matter of "formal" or "informal" use - the words are simply not frequently interchangeable.They are interchangeable in the restricted cirumstances in which 'elder' is used as an adjective (see post 2).
And yes, while you would properly say "She is older than her brother," only if she was out of earshot could you risk saying "She is the oldest" instead of "She is the eldest" without offence. Not necessarily. Some people do not use 'elder/eldest' at all.
With men, it's different. You may say, "He is my older brother" with a sense of pride attached to it. In fact, "He is my elder brother" sounds odd, as if you were trying to confer something (wisdom?) on him that's inappropriate in this context. Not necessarily.