Minor 6th chords use the major 6 interval off the root, just like major 6ths. The minor bit is the minor 3rd.A minor 6th chord with a minor 6th interval doesn't sound good. So 13ths will use the same 6th interval, but usually an octave higher.Strictly speaking, a 13th should have 1,3,5,7,9,11 and 13 in it, but that's often impractical - on guitar, for instance, there's only 6 strings, so all couldn't be played anyway. The other problem is that the order ought to be notes in numerical order. Fine on keyboards, but that's about all.Often, a 7th (usually flat) and a sixth with the basic triad will sound fine.
I don't think chords need to be made from modes, as they already exist in the 'mother' (Ionian) key.