In Galaxy Angel, it seems Mint and her family all have these droopy rabbit ears above their real ears.
I know that she likes cosplay so are these fake? In all of the anime seasons I don't see her without the rabbit ears and she seems to be able to move them as if they're real. I've read that the BROCCOLI game has some insight into the ears but I've never seen the game or could find any info about it.
It is unknown officially whether they're real or fake. As you mentioned, Mint's father, Darno, has these ears as well:
The reasons why Darno has these do not appear to be stated, nor are they very elaborate for Mint. It is believed that they are fake due to her losing them later in the series. From the Galaxy Angel Wiki:
She is notable for having a pair of rabbit ears above her human ears, a trait shared by her entire family and the family pet. They can move depending on her emotions and can flap rapidly for flight. Although generally considered real, they were shown to most likely be fake in the third season, after an ordeal with a duck helmet completely removed her hair and rabbit ears, while leaving her real ears attached.
So, while they are considered to be fake for the reason stated above, it has never been officially revealed one way or another.
Technically, they are テレパスファー (Terepasufaa, Telepasfer?), a peculiar parasitic creature on planet Blancmanche.
According to Mint's character introduction on Japanese Wikipedia,
In the original work, the parts that resemble animal ears on the head are parts of the body, organs for telepathy, and grow on all of her (Mint's) relatives regardless of their gender. In addition, GA2 made it public that the reason why Mint can use telepathy is the effect of Terepasufaa, a peculiar parasitic creature on planet Blancmanche. In the anime, they are ear attachments and react based on Mint's feeling. [...]. Also, just like Ranka, all of her family wear the same headband.
Galaxy Angel (original work)
(Mint) has a telepathic power and even able to read another, hidden, dark side of someone's thought, [...]
Galaxy Angel (anime)
(Mint) doesn't have a telepathic power, [...]
In the original work (game/manga/LN) for Galaxy Angel, it's only mentioned that the ears were parts of the body and capable of telepathy. However, Galaxy Angel II expanded the setting by introducing Terepasufaa.
Terepasufaa as first introduced in Galaxy Angel II: Zettai Ryōiki no Tobira (PS2)
In the game, it's described as a fluffy, parasitic creature living on planet Blancmanche. It doesn't attach to a human (most possibly, it only attaches to people from Blancmanche), and it rarely moves. It has a telepathic power and can be used by gripping it lightly, closing eyes, and thinking about people (unknown for other living things/non-living things). The user can see/know the targets' whereabouts & feelings, and even can suggest the targets to think about the user (to let the user know how they feel about the user).
Related video: Japanese Let's Play on NicoNico Douga (mirror for accountless viewing), starts at 0:33
A Japanese tweet by @mint_mamiya stated that Blancmanche conglomerates from generation to generation intentionally attach the terepasufaa to the newborn children. However, I can't confirm its reliability.
On the other hand, in the anime, they are just ear attachments. There's a scene in each season 3 & 4 where Mint doesn't put its animal ears.
Season 4, episode 26; image courtesy of Normad Type 3's Livedoor Blog (Japanese)
Note that the anime doesn't really follow the original work and thus there are some differences in the detail.
