i brought my girl to HKG when she was 5 months old:
Diapers wise: I brought half a pack there and bought some there. Diapers should be readily available in the major shopping markets
BFing: I was latching my girl on all her milk feeds and before i left the hotel in the morning i would empty my breasts after she feeds (my morning pumps are usually more than what she feeds) and pump after her last feed for the night for the same reason. I was moving between hotels and stored my EBM in the fridge to go so that if suddenly i got no milk, there's still some amount of milk
FM: I cant advise you cuz i dont feed her FM
Weather: Based on my understanding of Taiwan's weather June's quite okay in terms of weather, averaging about 26 to 28 degrees. But you have to be cautious of the rain (their so called rainy season) if you bring a stroller, please bring along a rain cover. if you are using a carrier, please make sure you have a cover/blanket to cover baby up if it rains. and umbrella too!