What Does it All Meeean??

So I was at my S/O house last night and he lives with two other guys who have cats. They left me alone in the house so I went wandering around and opened the basement door, forgetting thw cats were down there. So I happily occupied myself with the fur babies. I let them sniff my hand (I met them before but still) and the fluffy grey one approached me and started rubbing it's head all over me, and i began to pet it. It began to purr (never heard a cat purr in person) and eventually crawled into my lap to lay down. I cotinued to pet it and it continued to purr. But I have body language questions because when I was younger cats didn't like me, now that I'm older it's different and idk what to soo.

1) While on my lap, sometimes it would get into hunting mode (butt in the air), but it was facing a wall soooo what was that?

2) sometimes it would just reach out one paw and put it on the wall and keep it there while i pet it, what is that?

3) Sometimes it Would want me to pet it but pushing its head jnder my hand and I would per it on the head and then it would turn its butt towards me so i kinda pet the back area and it would turn around and look at me and ask me to pet it again with the same head gesture, what was that?

4) it kept rubbing its head against my phone?

5) Last but not least, i had closed the door because the guys came back, but they started playing the game so I went back to the cats and the grey one let me pet it again but at one point it would sit (normally just how cats sit, not in a threatening kind of way) and stare at me. Like I would distract it a couple of times but it always went right back to staring at me and it's pupils were huuuge. What was that??