I thought agasalho could also mean coat.
You're right, albeit "agasalho" is sort of old-fashioned. The sort of thing your grandmother would say. At least in EP.
or sobre tudo?
Yes, sobretudo, without the space.
Can this mean each person has a coat of their own or just that they all jointly own a coat?
The latter option is also applicable in this case.
Just 'Têm um casaco' is marked wrong. I cannot skip the pronoun this time?
why does tem have a roof on the e (ê)???