나타샤, you are correct. 제1화, 제1회. both are correct.
The difference is 제1화: '화' means story, episode. so it means '1st episode'.
'회' means the sequnece of event.
We can see both of them on drama.
By the way, '화' is only for the story, episode it used in darama,novel..
'회' is used generally in sequence things ,
Ex) 제 3회 동창회: 3rd skool reunion, they held the ceremony third times'
제116회차 복권 당첨 번호 on the lottery , (제)58회 정기 모임 on the meeting, 제 5회 결승전 on the game..et cetera.