@friendly @SveNo hello, PR of the series here! So I had the same questions when going through the lines and did some googling. The translation basically says non-melting cheese lol but it's probably cheese with a lower moisture content which could be cheddar. Based on that and the image in the manga I assumed it was just processed sliced cheese but I left it as that because I don't know enough about cheese to be specific.
About microwaves, it's common to use microwave ovens, that is a microwave and an oven in one appliance that is just the size of a microwave, in Japan. Even my home in Singapore has one. In this video from Rachel and Jun on YouTube
https://www.youtube.com/watch?v=lIIsZrbo4qQ, Rachel bakes banana bread in their microwave oven. I agree it was odd that they (the manga) put metal in the microwave but google says baking pans are generally made from a material that is safe to use in the microwave.
As for baking sheets, I assumed it was just another term for parchment paper? I apologize if they're two different things. They basically just need something that is non-stick to line the bottom of the pan with so the cheese doesn't stick to the pan.