This discussion may contain spoilers!
After the first playthrough I thought Faith was actually meeting the Woodsman because of the money he owes her. Then I played the episode again and noticed that she was wearing the ring right after she frisked him. She states several times that she wouldn't leave after she gets what's hers.
As you know Woody doesn't have any money. The truth is she was searching her ring as we know from the book that it belongs to her. The money was just a make-believe, Bigby wasn't supposed to know about the real reason Faith being there. The dialogue about giving her the money or not was just to make us also believe it's about the money.
When I mentioned this in another discussion @DomeWing333 pointed out that she didn't wear the ring in the room upstairs during the fight but she does outside BEFORE frisking him.
I checked it and it's true as you can see here:
You can clearly see she really doesn't wear it in the building but she DOES when she's slamming the axe into Woody head. Right after the frisking is done there is a relatively long close-up of Faith looking down at her ring thoughtfully. Is the ring in the last picture probably a glitch/oversight? I don't think so. She must've found the ring in Woody's apartment.
- Why would Woody have it?
- Why doesn't she want Bigby to know about it?
- Is the ring itself of importance?
- I can hardly remember Toad mentioning that he is keeping the left belongings of former tenants, that is why he had Faiths Donkeyskin. She seems to be a former tenant and I assume she once inhabited the Woodman's apartment. That would be an explanation why she approaches quickly to knock him over with the axe: she knew exactly where the ring is because she had hidden it herself earlier in the room in a place even Toad couldn't detect.
- I think it's just her who doesn't want Bigby to know about the ring or her identity because the ring is placed intentionally in the mouth of her chopped off head by someone else who doesn't bother.
- On the one hand the person who placed her head on the stairs wants Bigby to find out about Faith's identity and doesn't care about the ring itself as he gives it away into other peoples hands. On the other hand I remember Dee / Dum looking after something in Woody's apartment. When Toad told one of the twin brothers about the Donkeyskin coat Dee/Dum wasn't interested at all. I think they were looking for the ring and there are two groups of different interests such as a) Dee, Dum and their employer and b) the group/person responsible for placing the heads on the staircase
What do you think about it? Any Otter theories coming to your mind?
(Sorry for my bad english, it is not my mother tongue)
Proceeding with my idea of Faith being a former tenant of Woody's room: I think that one of the Tweedle twins was looking for the ring at Woody's place at Faiths former apartment while the other twin brother was looking for it at Lawrence's. If you go to Lawrence's first and hide in the cupboard you can see that Dee/Dum doesn't care for Lawrences body, he just sniffles around.