@Malorem TL checker here, you should probably get some facts straight:
It's not 玉子焼き. It's 目玉焼き. Go ahead and pop that into google and tell me what comes up.
Again, incorrect. Coughing wouldn't even make sense. And the same onomatopoeia can represent entirely different things (e.g. からから)
What she's jealous of is her ability to snap back to being a jokester even after being relentlessly chastised. 純粋 means pure or genuine, but *純粋に* means "genuinely"
https://thesaurus.weblio.jp/content/%E7%B4%94%E7%B2%8B%E3%81%AB
And to your last point, you're just nitpicking. つまみぐい means to sneak food you're not supposed to have, in this case, ingredients.