However, Australian/New Zealand Kmart stores have had no connection to American Kmart stores since the 90s, when Kmart Corporation sold them off to an Australian company that has managed them much more effectively.
One time while digging through a pile of 2x4s, an employee asked if I needed anything. I sarcastically asked what time they watered the lumber each day, so I could try to come get some before then. He just deadpan said "uh...I'm not sure."
It has the most bottom of the barrel staff and a lumber yard that sells... something... that barely qualifies as lumber. It is the epitome of modern American capitalism. It exists solely because it exists. No one would ever see a current Home Depot when visiting someone and say 'oh my god, please replace our current store(s) with this!'. But department stores used to offer experiences where people said that.
I don't think so. Home Depot is clearly a big hardware store / lumber yard. Costco may carry many kinds of merchandise, shelved roughly according to type, I don't think it really qualifies as it has unspecialized staff (and very few of them), little selection within each type of item, and probably its only true "department" is the tire department.
The only thing you list that could arguably be a department store is Walmart, and I think the reason it's not recognized as one isn't because of its size, but because it's been drained of all glamor.
And none of them (at least in their non "super" sizes) are subjectively much bigger than the old mall department stores.