The anoixantaria are indeed a musical setting of the last few verses of Psalm 103/104. They are (to my knowledge) chanted only in Plagal of the Fourth -- and only when a Hierarch is present.
A "terirem" -- I think that is a mis-spelling -- is said to be the sound the angels make. There is a whole elaborate Byzantine theology (of course!) of the terirem, but, musically speaking, it's a space filler: A chanter uses it in the long, fancy settings of hymns (like the Cherubic hymn) to play with the scale, show off and take up more time. In monastic settings, it just takes up time (gotta fill up those 9 hours for Vigil somehow!). I think when I was on Mt. Athos we sang various rems for about 35 mins straight.