Why is  Ice Cream used in Hot Buttered Rum?
I don't think I've ever even had hot buttered rum, but I can take a stab at some of your questions:


1 part ice cream and 2-3 parts boiling water will still average out pretty hot. If the ice cream is at, say, -20 to -10C, that'll get you in the 60-70C range initially, which is I think fairly normal for hot drinks. And of course, if you keep it in the refrigerator instead, your drink will be hotter!
I suspect mixing ice cream with butter will help it all mix together better in the end, by helping disperse the fat.
Ground spices aren't going to need a ton of steeping. I imagine the boiling water would be enough, and if not, the pre-cooking could easily compensate. And since some recipes just have nutmeg sprinkled on top, it sounds like steeping isn't necessarily the goal. If you're really aiming for spiced hot buttered rum, you'll probably want more spices than those recipes, and perhaps to cook them briefly with the butter.
If you intend to freeze the "batter", including ice cream will help keep it soft enough to scoop/slice. If it were just butter and sugar, it'd get quite hard in the freezer.
Yes, the milk/cream content of the ice cream would make the drink a little creamier - but I don't know whether hot buttered rum should be a bit creamy. It sounds like it'd be fine either way - personal preference, perhaps?