Gamanit
1)Assume nothing. Everything is in seforim. Moshiach will arrive first according to many Rishonim. According to Rambam he will in fact oversee the building of the 3rd BH. (He doesn’t hold like Rashi and other Rishonim who hold it will descend in fire)
2)The BH has a roof. The courtyard obviously does not. Check google images you’ll see.