If a user recovered data directory of LN node software with an old backup, what would happen? Would cooperative channel closing be possible? As far as I know, the "Data Loss Protection" feature would just request the counterparty to unilaterally close the channel. I wonder if mutual closing is still possible, with old channel states?

By the way, I roughly remember that I once seen a comment like this:

Before the "Static Channel Backup" feature was introduced, there was once a bad situation that the funds could be permanently lost, even if the channel counterparty unilaterally closed the channel. However, this bad situation was a trade-off in exchange for better (or easier) watchtower implementation.
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.