Seems like it was spelled that way looking at the etymology to differentiate kurb and curb. Seems like when horses were still prominent curb referred to a strap used on horses.

https://www.etymonline.com/word/kerb#etymonline_v_1818

Basically, curb was a type of restraint meant to hold back. So a curb/kurb on the street is just a tiny retaining wall holding back dirt. Some times it is fun to look at etymology. Also makes phrases "like curb your ambition" make more sense.

No, we call it a pavement

The “kerb” in the UK is the step between the road surface and the bit the pedestrians walk on. More properly “kerbstone” I guess.

Something to remember is that not very long ago, many words didn't have an accepted spelling, they were spelled phonetically. Although Curb would sound correct since it might be to curb things on the road from hitting the pavement/sidewalk, spelling it as kerb would have been OK.

Bill Bryson talks about this in one of his books (long time ago, I can't remember which) and it explains weird things like why the US pronounce Derby as it sounds but the Brits call it so it sounds like Darby.

Yes, and they also spell the things on your wheels "tyres"!

"Mind you don't scrape your tyres on the kerb!"