just to confirm

if we have 3 houses, 1 of each type, the prediction should be 1+2+3 = 6.

no , if you look closely at the sample submission for each image you have to identify the type so if an image is called imageone you will have imageone_{rooftype}then the count of that roof type

you're correct, didn't notice that the id is changed in the sample submission

What if you misassign roof type counts? Say for tin houses, you assay the counts for thatch houses and vice versa. Since the metric is MAE, does it affect the score?

yes it does