I've always wondered how/why a Redguard was made a housecarl in the notoriously xenophobic province of Skyrim. Any ideas what made Beth create her as a housecarl? Is there a specific reason behind it?

Probably because all housecarl in vanilla skyrim is nord so they try something different with heartfire, making a redguard housecarl.

As for game reason, Falkreath have a high elf priest, wood elf steward, redguard alchemist so I think they are quite open compared to other hold.

Ahh, so, not only are they the nearest hold to Hammerfell (one reason I guess), but they're also liberally minded and most multiracial?

The border with Hammerfell is quite close to Falkreath, and the city of Elinhir is just across that border. The two cities would have traded a lot - any time they weren't at war with each other!

because not everyone in Skyrim is racist or xenophobic.
A stormcloak Jarl has an Argonian housecarl. or is he a steward.

you can make housecarl backstories up really easily because they have no backstory at all. you can come up with your own reason for why Rayya is a housecarl.

Yes, that's one of the main reasons why my character chose Falkreath for where he'd eventually build Dragonia. It's got excellent links with the rest of Skyrim and the whole of Tamriel, along with being very hard to assault through those narrow valleys and heavy forests. It being beautiful there helps too!