Is entering "chat" prohibited (haram), disliked (makruh), or permissible (halal)? And does the one who opens it bear the sins of its users?

A chat is a gathering, and its ruling depends on what is said within it. If it is permissible, then it is permissible; and if it is forbidden, then it is forbidden. If it is used for da'wah (inviting to Islam) and teaching people, then it is desirable and merits reward.

As for creating a chat website, it is conditional upon the ability to control its content so that it is restricted to what is permissible. Otherwise, it is not permissible, and its owner is considered an accomplice to sin.