This is one of the paladin promotion quests. Talk to Grehgknak the Right in Lindisfarne Monastery.

... "rats in the basement" quest from the bank in Sturmford (there is a note on the ground indicating that the banker will pay for someone to remove them) ...

There is no access to any basement of the bank in Sturmford, so there is no quest attached to this. It may be that this was a quest that was never completed, or it may be just one of the many notes scattered around for general interest and amusement (which I think is more likely).

... the first door up the stairs in the Temple of Honk is "locked" in the same way as the first door in Drangheim Prison. ...

This door remains locked until Fre gives you the quest to retrieve the Golden Honk from the Temple of Honk. Do I really need to say where Fre is?