I disagree with just about everything you said.
"My son didn't sleep through the night until he was 3 months old and I've just heard other people say most of their babies slept through the night a lot sooner". that is insane. 3 months is very early to be sleeping through the night, 6 months is early to be sleeping through the night. Maybe those people are the ones who formula feed or let their newborns cry themselves to sleep and then brag that their baby sleeps through the night. Babies are not supposed to sleep through the night, it's a natural protection against SIDS.
"I was just wondering when I should let her cry for a while to make sure she's actually hungry, because I don't want her to get used to me feeding her at night every time she cries". How about never?? I am very against letting babies cry by themselves alone in a dark room. And even proponents of CIO would never advise you to let a 6 week old baby cry. You are her mother, do your job and feed her. My 6 and a half month old still nurses every 3 hours night and day.
"Do babies just learn how to sleep longer without eating or do i have to train her?". Just a word of caution about so-called "sleep training". Sleep training is nothing but various ways of ignoring your baby and letting them cry for varying lenghts of time. If you don't want the inconvenience of having your sleep interrupted at night then perhaps you should have thought twice about havng children.
"Just curious about what other people did". Personally, I feed my babies when they wake at night hungry.
"How manyoz during the day at each feeding would she eat for her not to need to eat at night?". THere is no set amount. She will sleep through the night when she is ready. That may be next week or it may be two years from now.
My story? My firsrt baby woke every 1-2 hours all night EVERY night for the first three months. I did my job as a parent and fed her when she woke up. By 4 months she was sleeping much longer stretches and by 6 months she was sleeping from 7pm til 7am. I did nothing to "sleep train" her, she did it on her own without us letting her cry ever.
My second baby, as I said, slept great until she reached 3 months. She is now 6 and a half months and still wakes every 3 hours to nurse. She has double pneumonia right now and has been feeding every 2 hours because she didn't eat for a few days. I will not sleep train her, at least not yet. I will see where we are at when she is one year and go from there. Right now, she clearly needs to eat at night so I feed her.