@MBloch's answer is Halachicly right. However, I and my Chevrutah have found an interesting theoretical distinction that must be made when approaching the Sugyah, not found in Halachic polemics on the subject:
Let's start with Mbloch's answer - all things attached go with the house - "The gold-plated faucets as would the built-in air conditioning unit". I'd ...
R Shlomo Riskin (the Chief Rabbi of Efrat in Israel) writes indeed that a ger toshav can own land in Eretz Israel
According to Rambam [Avodat Kochavim 10:6], the very term ger toshav, stranger-resident,
defines the fact that any gentile who accepts the seven Noahide laws
of morality may be a resident in the land of Israel, may purchase or
rent land ...
The Rambam (in Hilchot Mechira 25:5) brings the following halacha
When a person sells a house, he also sells the oven, the range, the
door frames that are attached with mortar, the door, the beam, the
lock, but not the key. [...] Nor does he sell the blocks for the feet of
a bed, nor the window frames, even though they are affixed with
mortar, for ...