1) I can't find the verb after "as would".
"Would" is verb enough. What that verb means is dependent on the modificand of the prepositional phrase that "as" creates:
staring at her as would a calm dog whose yard she'd walked into [stare at her]
- or -
staring at her as a calm dog whose yard she'd walked into would [stare at her]
We might consider this to be an ellipsis, or we might regard the auxiliary as anaphoric. Either way, the "missing" part is supplied by the phrase to which the entire prepositional phrase starting with "as" is attached.
The placement of "would" is flexible. Bringing it to the front of the clause helps to mark the clause as subjunctive. In this context, we have an actual seal and a hypothetical dog.
The embedded prepositional phrase "into whose yard she'd walked" isn't relevant to finding the antecedent of "would" (if we're calling it anaphoric) or its potentially redundant argument (if we're calling it elliptical). Rather, it is relevant to the other part of your question.
2) after the relative "whose", there should be a subject and verb.
a calm dog whose yard she'd walked into
This "yard" isn't the subject of anything.
Here, we have the modificand "a calm dog" and the relative clause "whose yard she'd walked into". In the clause, "[had] walked" is the verb, and "she" is the subject. "Into" is a preposition, and "whose yard" is the preposition's object. It is as if she had walked into a calm dog's yard.