I know in some religions and denominations, a building housing the officiant at a nearby place of worship has a special name, like rectory, parsonage, etc. Does a rabbi's place have a similar descriptor?
Generally not. As TrustMe mentioned, sometimes in the Hassidic world it can have a role.
Some synagogues do own a house reserved for the rabbi; they may refer to those as a "parsonage" for lack of a better term. It's not about the house being holy, it's very practically about the benefits package for the rabbi. (Especially as he can't drive to synagogue on Sabbath, and often it's an area with pricey real estate.)
I have heard the term used among Chasidim when referring to being in their Rabbi's home as being:
Within [the] holy abode [lit. inside]
Hope this is insightful!
No. There's no systematic, special name for a rabbi's house.
In some religions the institution owns a residence, which is used by that institution's (head or only) clergy person. While there are undoubtedly rabbinic homes like that, that are actually owned by a synagogue or other organization, all the rabbis I know bought or rented their homes on their own. (I wouldn't be surprised if Chabad owns houses, but I don't know.)
In the US, the tax law allows for some portion of a synagogue rabbi's compensation to be in the form of "parsonage" instead of salary. That money must be used for housing and has different tax treatment than the salary portion. I believe this is so that clergy of one religion won't be tax-privileged compared to clergy of another. Perhaps it is this "parsonage" that you're thinking of? (I don't know all the details; what I know comes from a conversation I once had with a rabbi about compensation structure.)