Air…I need air, and water. Water for my dry throat.
That last sentence feels a bit too tacked on - Koji is panicking here, so his thoughts should be simple and panicky.
Hot air from the nearby kiln entered my lungs, giving me breath once again—but the tightness in my chest remained.
How does "hot air" give him breath? I like the image this sentence evokes, but you seem to forget about that tightness right after and never refer to it again. Actually, Koji acts fairly normally from here on out, aside from taking a brief moment to think about his dream later.
A rosy hue spread across my cheeks as I self-consciously rubbed a sleeve across my dripping forehead.
Since this is written in first person and we assume Koji can't see his reflection, he wouldn't know that he was blushing. He could totally feel heat rising to his face, though! It would also be nice if we could see a bit more of his embarrassment here since his sister is making a pretty inappropriate joke at his expense.
I heard the sound of the front door creaking open down the hall, and a pair of tiny footsteps waddle into the house. My sister left and for the next few minutes, she and the visitor conversed in the kitchen—enough time for me to lie back on my bed and mull over the contents of my dream.
It had been the third night in a row I’d seen fire in my sleep.
"Tiny" is a visual descriptor - since he's talking about the sound of the footsteps, consider what "tiny" footsteps sound like - especially if they belong to webbed feet. Also, a "pair" of footsteps implies that there are two people - you would probably say "a set of footsteps" or something like that.
He also doesn't really do much mulling, here. Remember to show, don't tell
; show us what some of his thoughts are regarding the dream. He could be concerned, or intrigued, or simply confused and trying to figure out what made him dream of such a thing. Show us some of his thought process.
Emi entered the doorway clutching a kettle, and alongside her, a stubby, brown bird Pokémon wearing a happi coat covered in splinters.
I think you have too many commas here, and you're missing a verb for Fetch'd. You can change it to something like "alongside her was a stubby brown bird Pokémon."
Farfetch’d sniffed and turned his beak up at her, swiping a cup of tea.
Just wanted to say I love
this imagery. Also, I am intrigued to know how a Farfetch'd holds a cup.
Farfetch’d turned to me with a worried look. I bit my lip and avoided glancing at the Pokémon.
I might say "avoided making eye contact" simply because Koji already saw Fetch'd's expression, so he's already glanced at him. Eye contact is a bit more direct.
“It’s not fair for me to ask you to do this… It’s my responsibility, not yours.”
You don't need to capitalize words after ellipses.
I sighed and ran my hand through my dark, wet hair
Like I said, it really is hard to describe narrators within first person POV. The texture of his hair is tangible, so describing what it feels like works
(though I would emphasize that it's wet from his sweat), but people generally don't run their hands through their hair while reflecting on the colour of their hair.
I’d made the cups as a birthday present for my mother. The clay came from the bottom of Slowpoke Well, and I remembered spending a whole afternoon carefully molding it with my bare hands to make the perfect drinking cups. Now they were just some dishes lying on dirtied mats.
This provides a nice piece of background, though it does come across as a little bit too exposition-y. Maybe you could have him noticing the cups at the beginning of the scene, but still give this background info at this point? The way you did it kind of works because he is looking at the cups and thinking about them, but I'm not 100% sure that it works seamlessly. The last line definitely feels a little contrived - after all, they really are