Should one put a mezuzah on a doorway that is shared with a non-Jewish (or a Jewish-but-non-mezuzah-putting) neighbor?
If yes, should/must one ask their permission first? Should/must one tell them what one is doing?
It is always nice to ask. Particularly if your roommates won't know what it is. They could feel excluded if it is done but they don't know what it is or why it is there.
Should have a problem with it...well...this is just a sign of trouble to come. Nip that in the bud early to avoid long term problems.
Here are some relevant selections from Oholei Yeshurun, Vol. 2, Chapter 1: "Laws of Mezuzah," Section 1: "Laws Concerning the Ownership of a Dwelling," by R' Aaron Felder. Please note that while R' Felder is a talented and respected Posek who cites an impressive collection of sources, his is by no means the last word on the subject. If these questions come up practically, you should consult your Rabbi.
It sounds from this that according to R' Felder, the answer to your first question is "yes." I don't think I've ever seen a non-Jewish-owned apartment building in which the Jewish residents put a mezuza on the building entrance, so either 7a applies to Jewish owners, 7a isn't universally practiced, or everyone's afraid of their mezuzot getting damaged.
It seems to me that you're going to have to ask your Rabbi about your particular situation, to determine how ownership and fear of damage might apply.
To your second question, it seems a matter of common courtesy that if you determine that you need to put up a mezuza, you should let all affected parties know what you're doing.