(WIP)
Family Structure
Lineage/bloodlines is not important to Khajiit, possibly because Khajiit can bear kits from multiple fathers in the manner of cats, but also because the widely varying furstocks make tracing bloodlines through physical characteristics difficult. Emotional and/or spiritual ties is what forms a family instead--multiple parents and half-siblings all living together is not uncommon. In some parts of Elsweyr, families may tend to group up along the lines of the furstocks instead of by blood family.

Gender Roles? (uncertain of these)
Families tend to be matriarchal, with one Clanmother who "rules over" the other females and cubs. Males tend to roam (either alone or with other males) in hunting or mercenary parties, but some will settle with a favorite clanmother and/or mate and serve her clan.

Prostitution as a crime is not a crime to the Khajiit in the same way as it would be to other cultures. Females are considered to have reasons of their own for pursuing mates and thus are not questioned, and the males may or may not participate in raising any resulting cubs (as the clan will help raise them regardless). The crime comes in when a clansmother may be attempting to improve a certain bloodline or breed for more of a certain kind of furstock, in which case a female not following her orders may be branded as selfish and shamed. This is usually more of a problem anong the upper class.

Clanmothers
Tend to be the eldest female Khajiit in the community with children of her own. Passing on of the title may be based on number of kits (including grandkits) had? Or else by age/wisdom. Would have to learn math as part of their schooling, to be able to predict the moon cycles! Somewhat similar to wise women of the Ashlanders in terms of their role in society. Known for being embroiled in many plots at once, though these are ultimately for the good of their people, rather than any self-indulgent enjoyment (which is a cardinal sin to Khajiit).

Philosophy and Law
Khajiit see themselves as the secret protectors of Nirn, and have learned many tricks to hide their true purpose as well as out-manuever their enemies. They value adaptability, and no furstock is seen as better than any other, only different and best at different tasks.

Khajiit tend to be very insular from other races, and other races thus mostly know them as liars and thieves, as Khajiit will practice their plots on them, or use them for their own ends. Khajiit usually don't accept the validity of laws outside of their own, and make little of being dishonest to other races. Outright cruelty (murder, rape, slavery, etc) is still rarely condoned, as it is too direct a declaration of intent; a Khajiit disdains the method, but not necessarily the philosophy that drives the method. Meaning, they hate and disdain and disciriminate as much as the next Average Joe, but act on it in much more subtle ways. "A mob" is probably not a thing among Khajiit, except those very doped up on skooma.

Among themselves, Khajiit are very collectivist. Being too individualistic may be shamed for being too selfish, and so most pioneering make-a-name-for-themselves Khajiit have moved to other provinces.

Possessions are property of the entire clan (or sometimes more specifically the clansmother), and so "stealing" can generally only occur between two different clans. This is often a preferred way of settling disputes than through war (which is again, too obvious in intent), wherein stealing a prized possession of the other clan and flaunting it about, or otherwise making them the butt end of some trick or prank, shames the other clan and casts doubt on their fitness to be a protector of Nirn, and hence their "manhood".

^This may be a more recent development in Khajiit culture though, and before the Riddle'Thar, intense conflicts over resources, religious ideologies, etc, may have been more common.

Moon Sugar
Moon sugar is a method of gaining magical power (it is literally Magicka) and enlightenment, and used in many religious and pre-battle rituals. However, as it comes from the moons and is precious, a Khajiit using it for recreation or any other selfish purpose is shamed. A Khajiit in moon sugar withdrawal might be pampered and honored or alternatively shamed and abandoned, depending on why they were using the moon sugar to begin with. Skooma is typically considered an indulgence and aberration of moon sugar's holy properties, though Khajiit will also sell it legally to get the upper hand among outsider races. (For a trade deal, or to help an enemy succumb to their addiction, etc, etc.)

Religion
The Khajiit primarily worship different aspects of Jone and Jode, who they see as guardian spirits of the world, and which appear to be responsible for keeping the veil intact between Nirn and other planar realms. The Khajiit ingest moon sugar to gain their power, which they believe comes directly from the moons. The moons' phases "illuminate" the other deities, like light shone on two different sides of a statue makes the statue distinguisable from the shadows and more detailed and "3-D"--so the Lunar Lattice guides and structures the Khajiit's morality and ways of life. Most deities are thus connected to certain phases of the moons and/or the seasons.

Deities
Ahnurr (Anu) and Fadomai (Padomay) made the world by giving birth to a series of Aedra, Daedra, and finallyaspects of the world itself. Ahnurr is characterized to be stern, rigid, and jealous--the worst of Khajiit, while Fadomai is tricksy, motherly, and secretive--the best of Khajiit. In general, the Aedra are the First Children, and the Daedra the Second Children. The Last Children came last (obviously), when Fadomai tricked Ahnurr to mate with her one last time.

First Children
Alkosh (Akatosh)
Khenarthi (Kynareth)
Magrus (Magus/Sun)
Mara
S'Rendarr (Stendarr)

Second Children
Hermorah (Hermaeus Mora)
Hircine - Senche-Raht
Merrunz (Mehrunes Dagon)
Mafala (Mephala)
Sangiin (Sanguine)
Sheggorath (Sheogorath)

Last Children
Jone and Jode (Masser and Secunda)
Nirni (Nirn)
Azurah (Azura)
Lorkhaj (Lorkhan)

Nirni was a daughter of Fadomai and wanted to have many cubs of her own. Lorkhaj promised to send her to a secret place to protect her, but instead tricked and trapped her. Fadomai's other children punished Lorkhaj, and continue to aid or help oversee Nirni's children.

Azurah is considered a favored daugher of Fadomai, who she gave many of her secrets of Change to. Fadomai told Azurah to change some of the first mortals into Khajiit and make them into the protectors of the world. Y'ffer told Nirni this was a trick to steal her children from her, and helped her change the other mortals into Bosmer. Nirni favored these Bosmer and is constantly making life difficult for the Khajiit due to Azurah's supposed betrayal.

Other Religious Figures of Note:
the Dro-m'Athra
(de-retconning the ESO retcon here) Dro-m'Athra is a general term for entities that are "inverses". (Similar to that "Senche" is a term for anything big and bestial, though it also can mean a particular furstock.) Anything could be a dro-m'Athra in this sense, but it is usually taken to mean creatures from other worlds (such as the Daedra) or a corruption of the natural order (such as Namira's corruptions, or the Bent Cats of Lorkhaj). Similarly, the word dro-m'Athra by itself is not a value judgement (the labelled inverse is not inherently good or bad), but depends on the context in which it is used. "Dar-m'Athra" may be a term for Lorkhaj-specific Dro-m'Athra as seen in ESO.

Y'ffer
God of the Bosmer--though he helped stabilize the forms of the other mortals, he had no hand in the creation of the Khajiit. Tends to be considered an antagonist.

Namira
The Great Darkness that fills Lorkhaj--consumes from within, twists goodness into evil. Can also possess Khajiit, making them dro-m'Athra of themselves.

Rajhin
Cultural hero rather than a deity. Embodies many of the qualities of Khajiit--being a trickster and deceiver.

Riddle'Thar
Philosophy rather than a deity. Unites the Khajiit into one nation, tells them how to guard Nirni, tells them how to be adaptable and use their forms to their best advantage, likely tells them how to embrace change while also maintaining a kind of order (like the love-detachment false dichtomy of Buddhism).

Baan Dar
May be one and the same to Baar Dau (Lie Rock over Vivec). Signifies the defiance of the Khajiit, and helps teach them tricks to overcome their enemies.

Ius
Demi-god said to watch over animals.

Calendar System, Holidays
As a tropical climate, Elsweyr sees wet seasons and dry seasons. (In Earth terms, wet and dry seasons are caused by the earth tilting and so the rainbelt flip-flopping between different regions. While this could be Pellatine and Valenwood vs. Anequina and Hammerfall in Tamriel, I don't know if we want to metaphysic it up in a different way.)Thus, Khajiit in Elsweyr do not recognize winter, spring, etc, but instead use the moon cycles as First, Second, and Third/Last. These are tied loosely to the deities, where holidays of the First Cycle honors the First Children (Aedra), Second Cycle honors the Second Children (Daedra), and the Last or Third Cycle honors family and the world in general.

Summoning Dates: these are spread out willy-nilly on the calendar, but I am a fan of the notion that the Khajiit year begins in May (as this is closer to the time when most animals give birth than January); also, it puts Azura in the Last Cycle, as she really does serve as the endcap of the Khajiit origin myth. The middle part of the year also contains the fewest Daedra Summoning Dates, making it a little more sensible for the Aedric Cycle.