Ideas

Groggy Yet Possessive.

- Sleepy cuddles

- Zim mumbles "My human" and "Mine" in his sleep.

- He holds you with the utmost care and respect, albeit sleepy, but refuses to get up. He's scared that you'll leave forever if he lets you go.

(Light Spoiler Alert: especially after they start looking for a way home. 

- It's no surprise that Zim doesn't know when to stop. He holds them for hours, even after they've fallen asleep twice.

- He likes to nuzzle his face into their neck. The reason is partially that— at least in my story— Irkens used to mark their mates. 

Marking came in two forms, biting into your lifemate's neck and rubbing yourself on them to share scent. (Like cats.) 

He would do both, but he's hesitant to bite them. It's a big commitment in Irk, much like getting married. And between his repressed self-esteem issues, future abandonment issues, and the impact it would have on his reputation, he'd rather not. (For now, anyway.)

(It's the same reason why he's stealing their jackets. He likes their scent.)

- Y/N quickly finds out that Irkens can chirp and purr. Zim usually tries to be quiet when they show him affection. He doesn't want them to know he's enjoying it. But when they wake up before him, Zim can't put on an act. (Because he's unconscious.)

- Irkens are designed to be killing machines. Irkens are supposed to be quick, ready, sharp, and aware. All the things that you're not when you're asleep. But it's clear throughout the show and comics that Irkens like naps.

I think Irkens would have their senses elevated when asleep. That would allow them to be ready for a fight even in their most vulnerable state.

That also means that Zim can hear them even when he's asleep. He doesn't wake up because he doesn't see Y/N as a threat. (But try shouting at him and see if you don't get a pak leg through the chest.) My point is he likes them. 

It also means— bear with me, I know I'm going on for too long— that he chirps if they compliment him. And he'll whisper sweet phrases in Irken. (As sweet as Irkens can be, anyway.) They can't exactly tell the difference between the language and the noises, though. Irken sounds like a series of chirps and clicks. 

- Scratch the "He doesn't wake up 'cause they're not a threat." If they tell him they love him, he'll snap awake before the count of one. (He loves them too—or will, I haven't decided yet— but he won't admit it.)