Does Petruchio love Katharine?
It is difficult for modern Americans to understand the institution of marriage in Elizabethan times. First, Americans are romantic when it comes to love. We believe couples are fated to meet: a there is only "one true love." Also, we are usually economically independent of our parents when we get married, so there is much more freedom of choice. Lastly, we treat women with much more equality when it comes to decision-making and role in the family.
Elizabethans, most Europeans historically, have married for economic purposes: to secure status, maintain property and wealth. Women were considered property for the most part and given little choice by their fathers to marry whom they choose. Fathers were expected to marry the eldest daughter first, and they were married to men usually older than them.
If you've read any of the primary sources of the era, you will see much more inequality between the sexes than in Taming of the Shrew, as hard to believe as that is. Women were placed in 5 categories: Virgin, Quiet Woman, Good Wife, Wanton Woman, and the Unquiet Woman. The first three were good: they kept their mouths closed and their chastity closer. Even when married, they remained silent and coy. This is Bianca, the perfect daughter. The last two were considered equally evil: to be a wanton is to be sexually promiscuous. To be unquiet, like Kate, was considered as equally evil.
So, does Petruchio love Kate? Can a man love a woman that he places into such disparaging categories? Can a man even know a woman if society expects her to be silent, submissive, coy, deferential, subject to his every whim?
Shakespeare is being lightly satirical in Taming. He saw the inequality in marriage, having not been very good at it himself. He creates a modern woman
There's a lot of disagreement among scholars about whether Petruchio really loves Katharine. My own belief is that he does not.
It seems to me that Petruchio sees Katharine as a challenge. He sees her like some wild animal that needs to be properly broken and trained.
Over the course of the play, we do not really see love, in my opinion. Instead, we see Petruchio constantly forcing his will on Katharine in an attempt to make her act more like a "proper" wife.
Examples are when he wears rags to the wedding, when he forces her to agree with ridiculous things he says, and when he mistreats servants to force her to agree with him.
None of these seem like love to me.