Why isn't it meraa bhaii? Why is it mere? Brother is singular and masculine
Oblique case. Whenever the object is modified (मेरे भाई का घर, where का घर is modifying भाई) then it becomes मेरे.
Or is it the plural form out of respect for the brother?
Is there a difference between पुराना and बूढ़ा ?
पुराना is for objects.
बूढ़ा is for people, especially the elderly (and can be used for animals as well).
Vinay92 says: "You can use बड़ा for comparing ages for anyone and not just old people."
The corse conductor needs to add a suitable explanation as to why it is "mere bhai" and not "mera bhai"; and why not "mere bhai ke"
It's being explained in the tips and tricks. https://www.duolingo.com/skill/hi/Family/tips-and-notes