The starting *note* is irrelevant in pretty near any modal system - flip through any songbook and you'll find exceptions in seconds. The starting *chord* is more likely to tell you something - i.e. it's usual for folksongs to begin with a stressed note that is either the root, third or fifth.

Lots of folk tunes end on a note that is not the tonal centre (commonly the third in major-ish modes and the fifth in minor-ish ones). That didn't happen in chant, which is one of the reasons why folk music needs a different concept of "mode" from the liturgical one. (Chant also associates a fixed set of melodic formulas with each mode, as the Middle Eastern makams and Indian ragas do - folk music is less systemstic about this, though there are rudiments of it, like the three-note do-mi-do figure at the end of a lot of major-mode hornpipes).

This is an example where the final note is not on the tonal centre (lullabies usually have tonally indefinite endings like this, so as not to wake the baby up by conveying that the song's finished). It would be appropriate to use a C chord at the end, i.e. the dominant of F.

  X:1
  T:I Know Where I'm Going
  M:2/4
  L:1/8
  Q:1/4=80
  K:F
       F F G A|F F2
  C   |A A A B|A G3 |
       c c c c|c F2
  F/F/|G A B A|A G3|]

The mediaeval modal systems had extremely practical ends. For the church, they let you organize a group of melodies for a single liturgy in a common mode (or a few related ones) so the choir would be able to sing the whole service in tune (almost literally "singing from the same hymnsheet"). It didn't have anything specifically to do with the use of notation. For the secular music of the Middle East at the same time, the modal system would get a group of performers in tune for a collectively semi-improvised piece - harps were commonly used a the time and they could not be retuned on the fly, you needed to set the mode once and for all at the start of the concert (a practice still followed in concerts of classical Turkish music).

Both systems had the same antecedents, probably in the music of ancient Mesopotamia, with its earliest surviving theoretical description by the ancient Greeks and its first use in the church in the Byzantine "tonoi" (which predate the Western modes by centuries and are still in use).