A roofless structure:
A house which does not have a roof is exempt from having a Mezuzah. If it is partially roofed, then if the roof is by the area of the opening to the room, and the roof is a space of at least 4 by 4 Amos, then it is obligated to have a Mezuzah.
A roofless room-Is the inner room of a house that does not have a roof obligated in a Mezuzah?
Yes. One is to place a Mezuzah with a blessing.
Is the entrance to a balcony obligated to have a Mezuzah?
Is the entrance to a backyard or garden obligated to have a Mezuzah?
If it leads to a home then it is obligated to have a Mezuzah.
Is the gate of a house with Tzuras Pesach obligated to have a Mezuzah?
 Michaber 286/14
 This is not similar to a courtyard or gate entrance of a city, which is obligated in Mezuzah despite not being roofed, as it nevertheless leads into an area of dwelling, such as to the homes of the city, as explained in the next Q&A!
 Pischeiy Shearim 286/170
 The reason: As the room leads to a room that has a roof and is obligated in Mezuzah. [ibid] This is similar to a gated courtyard that is obligated to have a Mezuzah due to it leading to a home, even though it does not have a roof.
 Minchas Yitzchak 1/8
 The reason: As it leads to a home and is hence similar to courtyard.
 Minchas Yitzchak ibid