Learning 7th chords by ear is just a matter of being able to hear the individual notes, so it's better to start with arpeggios where the notes are played separately. Here are some tips on how to identify these arpeggios.

Use intervals! If you haven't practiced interval ear training, you will be at a major disadvantage with this exercise. The first thing I listen for is whether or not the arpeggio begins with a major third or a minor third. If it begins with a major 3rd, it will either be a major 7th chord or a dominant 7th chord. Even with the notes played separately, a dominant 7th should feel a lot more tense than a major 7th. If it begins with a minor third, it will either be one of the diminished chords or a minor 7th chord. Minor 7th will sound quite bright, because the upper three intervals spell out a major triad. A fully diminished 7th chord will be consecutive minor 3rd all of the way up, so if you hear a sudden leap of a major 3rd at the top of the chord, that makes it a fully diminished 7th.

What's the difference between a half-diminished 7th chord and a minor 7th flat 5 chord? They are spelled with the same intervals, so for these exercise you can think of them of as the same thing. If you want to get technical, jazz musicians tend to refer to them minor 7th flat 5 chords and classical musicians tend to refer to them as half-diminished 7th chords. The function of a chord can change slightly depending on the musical genre, so maybe they are technically different things, but at this point we're splitting hairs.

It's also worth noting that if you harmonize the first note of a major scale, you get a major 7th chord. If you harmonize the tonic of a natural minor scale, you will get a minor 7th chord. Because of this, either of these chords can sound relatively stable while all of the others will feel the need to resolve. These tendencies are more obvious when the chords are not arpeggiated, although your ear can still be trained to pick up on these things regardless.

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