So I've started a Khajit whose goal in life is to give Khajits a better name: no melee weapons, no stealing or pickpocketing or murder, etc. She's facing an ethical question of whether to exercise a special grant to buy a house while other Khajits are prohibited from entering a city.

Her moral code is leaning toward saying 'no thanks' to the purchase offer unless the rest of her kind gets a general grant of rights. Which probably means marriage is her only route to home ownership.

Does anyone know of quests that, generally speaking, better the lot of the Khajits of Skyrim?

There aren't a whole heck of a lot of Khajiit in Skyrim to begin with, and it's only the caravans that aren't allowed into the cities.

As for quests that help them out... There's one that goes counter to your plans but helps the caravans make more money by throwing in with the Thieves' Guild.

That's about it though.

Nope.
I make friends with them, try to support them but it's tough...
(You can actually do a great deal to help the Argonians in Windhelm though.)
But a Khajiit dragonborn, eh? Wow... good for you.

Hey just playing today, I found a very unusual book I'd never seen before, all about Khajiits and their homeland and goals.
Seems they will NEVER rest until they get their lands back from the Count of Leyawiin...
Apparently he's Khajiit public enemy number one.

Nords will never understand that moonsugar is the equivalent of mead to Khajiits in their culture, so why would you bother.

Khajit get a bad rap in the series. In Morrowind they were slaves; everywhere else they're known as thieves and smugglers. So it seemed like an interesting roleplay choice to play a khajit as a redeemer.

Yeah, the marketing suits are clearly "dog people". I do what I can which the only thing you can do is get that amulet back for their guard. ...and I buy stuff from them every time I see them even though I don't need anything.

You have your work cut out for you trying to balance my Champion of Namira Khajiits dirty deeds and penchant for human flesh.
He even disgust Eola sometimes.

It's mainly that their culture doesn't mesh well with imperialized standards. It's changed over the years and gotten a little more imperialized itself, but it's still not good enough for most.

Also they have fur... we have a hard enough time maintaining tolerance between people with different colour skin in some places in the world. In terms of DNA that human to human difference is less than 1%. In terms of human to beast genetics that difference is about 2%. Add into the fold that they are a minority group does not help.