Hmmm...on the push, the air is being pushed outwards and the note might sound louder. Other than that, whether using the push or draw to start the tune will depend on the key and the initial note or scale degree of that note. G tunes will usually start on G (tonic) or D (dominant). The G is usually played on the push (left hand) but the D can be either hand and will depend on the octave. For the key of D, the low D will most times occur on the draw, as will the higher D on the right hand.
Regarding the violin, I play classical and trad and while the downbow is important in classical to emphasize the strong beats of the measure, this is not necessarily the case in trad music (which is what is usually being played with Anglo concertinas.) In Irish trad (and Scottish as well), jigs use a slurred bowing across the strong beats, even while the 1 and 4 sub-beats might be accented. This gives a very pleasing “swing”. Getting this same rhythm on the Anglo is very key to the “trad” feel. This swing is very evident in Mary MacNamara’s playing, and in the East Clare style in general.
My best advice is to listen, and especially to the Clare players, East and West. Try some of the lessons on the Online Academy of Irish Music and try to find a teacher, live or Skype.
Cheers