If a user recovered data directory of LN node software with an old backup, what would happen? Would cooperative channel closing be possible?
No, a mutual close is not possible. For signing a mutual close, both the parties should be aware as to what the current state of the channel is at. Since one of the party has fallen behind, it does not know how much it is owed (or more correctly cannot prove to the other party how much it is owed). Moreover, if the last signed commitment transaction included adding of HTLCs, it is impossible for the fallen behind node to know what those HTLCs are, if the pre-images were sent, or if the HTLCs have timed out.

The one way that can be though of is that when the other party detects that its peer has fallen behind, it sends its peer the last signed serialized commitment transaction (without signatures) that it holds and the signature that the fallen behind peer sent it. They can then negotiate the closing condition including the fees. However, the protocol allows only messages that are not more than 65535 bytes long. Commitment transactions can be as much as 400,000 WU (>100,000 bytes) so it would not fit in a single Lightning message. May be the protocol can use the new TLV format to indicate multiple messages to follow, but as is, the current specs do not allow it.