Shouldn't the cream be poured in before the coffee?

Yes, I do that at home.

But stirring or just walking to work with a cup in hand mixes it up easily enough.

In Europe, in my experience, that is done normally. The US seems to be backwards on this.

Sugar, as well. In the middle east (my experience) the sugar goes in first and then the coffee.

I put the sugar in first, prevents splashing.
As for milk/coffee, it makes no difference to me.

That’s the way I do it. It is the way I grew up drinking tea with my parents; sugar, milk (or cream) then tea. Mom was, and still is in many ways, old school. First thing you did when you walked through the door or when company arrived was put on a pot of tea. That and I am too lazy to stir.

If you put cream in first, how do you know there will be enough room in the cup for the right amount of coffee? I like about 10:1 coffee:cream, so if I mispoured the cream, I would need 10x the amount of coffee to balance.

I get around that problem by only filling the coffee mug half way. That gives plenty of room to compensate for a cream pouring error.