Pets, which house?
If we go by the owner's view. it's possible to argue pets are:
Sixth house - a pet (small animals only)
Seventh house - a partner
Eleventh house - a friend
Third house - a sibling
Fifth house - a child
Fourth house - a treasure; or even
Tenth house - the key social role is ownership and care
All of these can be sincerely held by the person and indeed could be recognised by others who know them. I can therefore see why some Astrologers would argue for them.
The problem for me is that none of them describe a 'real' relationship between human and pet. Most of these relationships are two way - can someone you regard as a friend actually be a friend if they don't see you in the same light. We can't know for certain that the relationship is a 'genuine' one, if we can't know the pet's feelings.
It can be argued that the pet's feelings are immaterial, it is the feelings of the querent that matters but even between humans this can cause trouble for the Astrologer when the feelings are one sided. There's a good instance of this in a horary by Lee Lehman, where the querent saw the relationship as a long term partnership but the other party did not.
I'm not saying that it's wrong to use them, but it may not be effective to use them. Certainly if you open them up for use the possibilities multiply almost to the point where horary becomes useless. And what does the Astrologer do if the querent says 'The dog is like a son to me?' but from observation, the Astrologer feels that the dog is more a pet? Should the Astrologer use the fifth, which they don't believe to be the correct house or should they use the sixth? It's the job of the Astrologer to tease out the 'correct' house for them to use to give an effective reading ('correct' here is not an absolute but for the specific question')
Of the relationships listed above only two are ones that could clearly be said to apply, another is debatable but may be true on investigation. Those are:
Sixth house - a small animal
Fourth house - a treasure (though even that might be debated)
The one that could be argued is Tenth house - the person's defining social role - there might be a case for that if the person earns his or her living from the animal performing or is known in their society simply in terms of the animal and has no other social role.
Traditionally, the Astrologer would use sixth house and possibly the fourth house if they believed the animal was also the person's treasure (it doesn't have to have monetary value and it's a lot less of a subjective judgement than 'friend' or 'child').
In Willowfox's original chart, using the fourth doesn't alter matters much - direction changes to be more South than West - Mercury would be Lord 4 and is placed in Scorpio ninth, so the conclusion would be that he is 'far away' and Mercury is stationary in a fixed sign - so Bird is not moving. Worse still the Taurus Moon is applying to an opposition to Mercury, so unlike using the Sun, Lilly's test for life would be failed.
I did consider this when I looked at the chart but I preferred the slightly more optimistic reading using the usual significator of a pet.
Incidentally, Mercury is also Lord 7, so if Bird were a partner, the same conclusions would be reached. Lord 11 is Saturn, also placed in the eighth, but in Scorpio. Lord 5 is the Moon, giving a direction of North East. Lord 3 is Venus giving a direction of due South. Lord 10 is Jupiter, which was assigned to Willowfox, so we either use Venus on the MC (due South) or the Triplicity ruler for fire - the Sun.
As Willowfox is the one that has to do the searching and knows the area, he might feel one of the alternative directions is more appropriate and I don't think in this case we can in any way override his own instincts
It's a heartbreaking situation, I know I would continue looking but horary is not a powerful tool in cases of moving animals, nor for that matter can it lead to the rediscovery of stationary animals or things when their location is inaccessible.