For the ship questions: Promptis!
Who said “I love you” first: Noctis. Prompto wanted to be the one to do it but he never felt sure enough because Noctis is the prince (!!!!) and Prompto is well... Prompto. So Noctis finally says it (with encouragement from Ignis), and Prompto stands there in shock before Gladio taps his back and he just ends up kissing Noctis
Who would have the other’s picture as their phone background: They both do, but Prompto was the first one to do it. It’s a photo of them together at the arcade.
Who leaves notes written in fog on the bathroom mirror: Prompto. Although Ignis sometimes leaves his own notes like “Eat your veggies!” But Noctis can tell the difference between Prompto’s sappy love notes and Ignis’s (they don’t tell Ignis that, though).
Who buys the other cheesy gifts: Noctis. Usually it’s because he’ll pass by something that reminds him of Prompto- typically it’s chocobo related. He gets it for him because Prompto loves getting gifts but will never admit to it.
Who initiated the first kiss: Prompto. (Kind of answered this one above.) He’s so overwhelmed by Noct’s confession that he just kisses him.
Who kisses the other awake in the morning: Prompto. Noctis sleeps in any time he can. Prompto is always up early after a morning run with Gladio (and sometimes Ignis).
Who starts tickle fights: Depends on the occasion. If Noctis is losing in a game? Definitely Noctis. If Prompto finds Noctis lost scrolling on his phone and wants attention? Definitely Prompto.
Who asks who if they can join the other in the shower: Noctis. It’s not so much a question as it is Noctis just telling Prompto he’s dirty and needs a shower and jumps in with him. Prompto lets out a lot of yelps and Noctis doesn’t even realize what he’s done until it’s too late.
Who surprises the other in the middle of the day at work with lunch: Prompto. Noctis gets overwhelmed with being THE PRINCE so Prompto often catches him when he’s on lunch just to sneak away for a few minutes.
Who was nervous and shy on the first date: They both were. Prompto was incredibly nervous and shy to the point it was hard for him to talk. Noctis was so nervous that he kept saying awkward things. The moment they said to just act like they “were friends” the tension released.... Until Noctis kissed Prompto. (Because they’re definitely not just friends)
Who kills/takes out the spiders: Prompto. Noctis originally wanted to just kill them but Prompto was NOT having it. “They kill other bad bugs!” Prompto takes them out now.
Who loudly proclaims their love when they’re drunk: Prompto. They don’t often get drunk, but Noctis is a sleepy drunk and Prompto is a mushy romantic when drunk. He just tells everyone how much he loves the prince, and everyone thinks he’s just really loyal to Lucis. (They’re not wrong.)