I believe the title refers to Kate and her efforts to capture Marlow's heart. Her father has selected her husband and neither of them has even seen him. Kate is afraid she won't like him because of the way her father describes him. In order to know Marlow's true feelings, Kate pretends to be a barmaid. When Marlow is willing to go against society and annnounce his love for Kate even though she is not of his class, Kate knows Marlow is a good trustworthy person. In order for Kate to " conquer" Marlow's heart, she must "stoop" by pretending to be a barmaid.
Marlow is rude at the Hardcastle home because he believes Mr. Hardcastle is a simple innkeeper and not a gentlemen. Tony has led Marlow to believe this. Since Marlow believes Mr. Hardcastle isn't of his class, he treats him rudely as a member of the working class.