My toddler has climbed me, laid across my shoulders like a cat, and gone to sleep.
Is this normal? Can it possibly be comfortable? (It isn’t for me.) Is my darling child part feline? I am baffled.
Across your shoulder sounds a little out of the normal but my daughter use to fall asleep with her head on my shoulder. Made it easier to eventually carry her to her crib/bed.
DemonTree? More like a climbing tree! 
I think toddlers will do all this and more. Mine specialized in other oddities such as sucking his mommy’s chin, and tearing my glasses off my face.
We went camping once, he lay in the middle between us, while we shivered all night with our backs pressed up against our respective sides of the tent. In the morning, there came a moment when he woke up and reached out, tweaking both our noses at the same instant. Hard. Not sure how he did that.
When my nephew was eight months old or so, he had a weird habit of sucking on my nose, sometimes holding my ears to anchor himself. I thought it was weird, but had no problem if that’s what he wanted to do. Twenty years later, it occurred to me why he did it. Just to be weird, I would give him his pacifier by holding the handle end between my lips and letting him take it from there. He apparently thought that his pacifier was somewhere on my lower face and my nose is the most prominent thing there. He did have a weird expression before he’d put his mouth on my nose, which I now realize was him looking for the pacifier. So the whole thing was my fault, because I didn’t know about what babies understand and don’t understand about object permanence.
My daughter likes to suck on people’s noses too, so it probably wasn’t your fault at all. Babies are just weird.
Lol, that’s impressive coordination!