Let’s start with your first question. The use of part of the room as a home office would grant it a distinct status as an independent room. Even if it is only enclosed by the accordion door from time to time, its doorway needs a mezuzah.[1]
To answer your second question, the railing is deemed to be a lintel for the doorway.[2]
Your third question is a matter of controversy. In fact, some authorities hold that one should hire a carpenter to construct a special doorpost for the mezuzah.[3]
However, many authorities hold that the fixed part of the door would be considered as a doorpost and the mezuzah should be placed there.[A]
Unfortunately, the mezuzah may then be facing out and not under the lintel. This is not ideal, but the best under the circumstances.[4]
Other authorities allow the mezuzah to be placed on the doorpost, even though it will then be more than a tefach away from the opening.[5][B]
In either case, no berachah should be recited on this placement.[6]
[1] Agur B’ohalecha 25:9.
[2] Leket Shut Mezuzah 2:2, in the name of R. Nissim Karelitz. See Sha’arei HaMezuzah 11:8.
[3] Chovas HaDar 8:3.
[4] Agur B’ohalecha 22:11; Teshuvos V’Hanhagos 3:328.
[5] Sha’arei HaMezuzah, citing Kovetz Mi-Beit Levi 3:5753; Ohalei Yeshurun, Mezuzah 7, citing R. Moshe Feinstein.
[6] Each one of the three issues raised in the question is a halachic controversy.