1) Does it sound odd to day "What should I set the air fryer's timer for?" or "What (time) should I set the air fryer timer for?" Neither? What about "What time should I set the microwave timer for?"?

I just assumed that an "air fryer" was some cooking device different from a microwave. If you used both terms for the same device, then Alexandr is right and you should say "microwave." Although, isn't there some kind of cooking device that will make fried potatoes just using convection heat? I'd call that an "air fryer," similar to an "air popper" used to make popcorn without oil.

1) Does it sound odd to day "What should I set the air fryer's timer for?" or "What (time) should I set the air fryer timer for?" Neither? What about "What time should I set the microwave timer for?"?

All of your sentences are acceptable. They aren't great English because you end each sentence with a preposition, but "dangling prepositions" are now widely used and acceptable in spoken English. I probably would just say "how should I set this timer?"

2) Why do you say "My car's tyre" but "My shirt sleeve" and no "my shirt's sleeve"?

I don't say "my car's tyre," I say "tire" because I'm American, not English. The difference in spelling may not influence the pronunciation, but you can be sure that I would pronounce the word differently than would someone from England.

Actually, you can also say "my car tire" instead of "my car's tire." In this case, and in the case of "my shirt sleeve," the words "car" and "sleeve" are being used as adjectives. In the case of "car's" and "shirt's" the words are possessive nouns.

So, all your sentences are acceptable. In fact, you can say "my shirt's sleeve," even if it is just less common usage.

What time should I set the microwave timer for? Is better. Also just forget the air fryer, cause you can't fry air.

About that Genitive case. I use to replace it with something else, if I'm not sure. Like u can say the tire of my car, or the sleeve of my shirt.

I also met car tires. If two nouns go one by one-the first one serves as an adjective.

Like spring (adjective) rain(noun)