Hi Tay,
No, he does not fall asleep, he simply stops sucking and pushes the nipple out, or sometimes holds the milk in his mouth.
Yes, I have him weighed every week. Wk 1 he was 3kg, wk 2- 3.4kg, wk 3-3.5kg and just this morning 3.7kgs.
He does have about 3 soaking wet diapers during day and 2 soaked ones at night...soothing (soaked as in diaper is heavy and sometimes soaps thru his clothes at night)