Lightning components are case-sensitive. So you should use exact field names when you access any variables.
You have Name as field of Attachment in apex class but you are trying to access name in your lightning component. Change it to Name, I think you should be good to go.
This is an issue and initially this seems to be the only culprit, if the issue still persists, please comment I will update my answer accordingly.
UPDATE 1: If you are able to fetch the Attachment records using inner query, you can access them like below:
<aura:attribute name="case" type="Case"/> <!-- Save the whole case record here -->
{!v.case.Attachments[0].Name}