Is it permissible to propose marriage to a widow whose husband passed away five months ago and who has three young children, knowing that the husband is already married and has children, and that his current wife will reject this marriage, and is this marriage an injustice to the current wife and children?

There is no harm in communicating with the widow to check on her well-being. Marrying her is permissible if you are able to be just, and the consent of the first wife is not a condition. However, if you fear problems with your first wife, it is better to refrain from marrying her and instead help her and her orphans through your (first) wife.