What role does the cat play in Beetle's first weeks in the village in The Midwife's Apprentice?

The cat plays a number of roles in Beetle's first weeks in the village.  Its existence is first of all a kind of mirror of Beetle's.  Both are homeless and on their own, and both are tormented, especially by the urchins of the town.  Until the boys try to drown him, he arguably does better in the art of survival than Beetle.

The cat is the first being with whom Beetle forms a positive relationship.  At first, Beetle simply watches him and sometimes shares her meager stores with him.  After she rescues him from the boys' cruel prank, she nurses him in the only way she knows how, wrapping him and keeping him warm, which she seems to have known to do by instinct, and cursing him, which is the only kind of spoken interaction she herself has ever experienced.  When the cat revives, Beetle and he begin to develop a quiet companionship.  She sits with him and shares what she can with him, and she tells him "what she could remember of her life before they found each other" (Chapter 2).

There is one more important role the cat plays during Beetle's first weeks in the village.  As Beetle watches the Midwife deliver the baby of Kate the weaver's daughter, it reminds her of when she pulled the cat out of the bag in which the boys threw him in the pond.  The memory of saving the cat enables Beetle to form a connection, and she understands that midwifery is actually a practical art, "as much about hard work and good sense and comfrey tonic as spells and magic".  With this knowledge, Beetle is able, through observation, to develop her own skills in midwifery, skills that will eventually surpass those of the Midwife herself (Chapter 3).