Present participle: so in vers. 56,57,58. It is characteristic: he is the 'eater' of this divine food.
Dia with the accusative is not simply 'by' or 'through,' and here it is evident that it is important to be accurate. The sense is 'by reason of what the Father is and his living;' 'I live by reason of his being and living.'
Not simply the act, but the will to do it: 'Is it your will or disposition?' 'Do ye also wish to go away?'