Is it permissible to chat with unknown women via Facebook, including young and old, to discuss their problems and try to help solve them? Should one completely avoid them, or continue to do so with a promise not to abandon them? And how can one expiate this promise if avoidance becomes obligatory?

It appears that the conversation has gone beyond the need for anything resembling friendship, and this contains elements of evil and temptation. The fact that the questioner did not pledge does not negate the danger of temptation.

The correct approach is to stay away from them. Fulfilling a promise is not obligatory according to the majority of scholars, and there is no need to fulfill the aforementioned promise. Rather, even if it were an oath or a vow, it would be better to break it, as the Prophet (peace and blessings be upon him) said: "By Allah, if Allah wills, I will not swear an oath and then see something better than it, except that I will expiate for my oath and do that which is better."