“When a boy courts a girl, he stands outside her house and sings to her.”
I looked the word “court” up in the dictionary, it says if a man courts a woman, he spends time being nice to her because he hopes to marry her.
Do people still use the word “court” in this kind of sentence these days? Does it sound old-fashioned?
Definitely old fashioned.
But kind of classy at the same time, if you were to use it at the right time. 
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Agreed. I often use it when I want to sound classy or old fashioned. Or both
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Agree, a bit old-fashioned and possibly classy.
Not old-fashioned, though, in other related contexts. Some real examples found in headlines from the web:
“Andre Iguodala being courted by several teams.” Several teams are trying to attract Mr. Iguadola so that he might decide to sign a contract with them.
“Ally McBeal First Courted Controversy 20 Years Ago.” That TV show attracted controversy.
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I would say it’s a useful word that can be used in many contexts, it’s just that the behavior that it originally describes has in and of itself become old-fashioned in some cultures. Swiping on an iPhone hardly qualifies as courtship, but courtship would be the proper word to use in other instances.
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How about the use of the verb “courtiser” in French?