Hi. I am a native speaker of Korean but also read IPA.
What about [y] and [ø]? Does anybody still use these vowels, or is this feature completely dead?
As people have pointed out, technically ㅟ is [y], and ㅚ is [ø], but almost no native speaker is even aware of this, let alone pronounce them that way.
In fact, when learning languages with these vowel sounds, like French, German, and Turkish, most Koreans have extreme difficulty (which is ironic because Korean technically has those sounds).
As far as I know it's supposed to be different than ㅙ and ㅞ, but when my friend pronounces all three in a row, I hear them as extremely similar. The only difference I can pick up is slight raising/lowering. At least comparing ㅙ and ㅚ. ㅞ, of course is different since you have the [ɛ]/[e] distinction.
As for the distinction between ㅙ, ㅞ, and ㅚ, I recommend that you just pronounce them the same: [wɛ̝] (or any variation of [e] or [ɛ] that is comfortable for you)
I assume that your Korean friend is 100% native Seoul-speaker. When I ask Seoul people to pronounce certain words like 개 and 게, I can sometimes tell the difference between the [ɛ̝] and [e̞]. But this is only when I specifically ask them to pronounce them side by side. Otherwise, they usually don't distinguish them either.
Although I am not from Seoul (my family is from the southeastern part), I still speak standard Korean. Still, I could NEVER distinguish ㅙ, ㅞ, and ㅚ in my speech.
BTW, [ɰi] for ㅢ is essentially a spelling pronunciation. In Seoul colloquial, it's either realised as [ɯ] (initially) or [i] (in other positions), with the exception of the particle 의, which is [e].
Here are the technical rules regarding ㅢ (which most native speakers are not even aware of):
1) 의 at the beginning of the word must be pronounced as the original ㅢ [ɰi].
EG) '의자' must be pronounced [ɰidʑɐ].
(Some older people of certain regions pronounce them [ɰ], but this is wrong according to the standard. And it really does sound bad)
2) 의 in the middle of a word, with NO consonant in front, can be prounced as EITHER ㅢ [ɰi] OR ㅣ [i].
EG) '하의' can be pronounced either [hɐɰi] or [hɐi].
3) Consonant + ㅢ/의 must be prounced as C+ㅣ [(consonant)i], WITHOUT the [ɰ] sound.
EG) '무늬' is pronounced [muni], NOT [munɰi].
also, '악의' (note that the consonant before the 의 also counts) is pronounced [ɐgi], NOT [ɐgɰi].
This means that '악의' (ill will/malicious intent) has the same pronouncation as '아기'.
4) Finally, as pointed out earlier, the particle 의 CAN be (and in reality, most of the time is) prounounced as 에 [e̞].