who cooks normally?: Basil, I'd imagine Death doesn't exactly know how to cook and would end up accidentally burning the house down or something
how often do they fight?: Not to often, occasionally Death will get upset at Basil's lack of self confidence but these fights wouldn't last too long
what do they do when they’re away from each other?: Death would normally be out collecting souls and Basil would paint, per usual.
nicknames for each other?: I don't think Basil would exactly be comfortable with referring to death as 'Death' though I don't think he would use a nickname, he would just avoid it all together. Though he calls him a grim reaper to tease him occasionally
who is more likely to pay for dinner?: Death
who steals the covers at night?: Basil
what would they get each other for gifts?: Basil would make him paintings or sketches while Death would probably go get him paint supplies and things to serve as inspiration like flowers and whatnot
who kissed who first?: Death kissed Basil first I'd imagine
who made the first move?: Probs Death, Basil is far to awkward to attempt to make the first move
who remembers things?: Death
who started the relationship?: No one strictly started it. It just sort of happened.
what would they do if the other one was hurt?: Death would probably freak out if anything happened to Basil and Basil would probably do the same, he doesn't think any harm can come to Death.