Part of that seeking is the establishing of strict cultural borders between 'them' and 'us'. The Uighurs present a significant insurgency problem to Kyrgyzstan and Kazakhstan too.
Dunno. Kazakhstan's main threats so far have been (entirely alien to the turkic tradition) Wahhabi Islam (KZ now regrets letting Saudi Arabia build so many schools, hospitals and mosques), and Russian bullying and sedition.
The president's clique is resented by some of the other big tribal factions/families, but thats a straightforward power thing, rather than any sort of national separatism.
Most of the other factions seem ok with being part of a proper central asian nation (especially one that has some oil wealth to spread around). Even most of the "white" Kazakhs (Russian, belorussians, baltics, volga germans, Ukrainians, etc - all former transportees of the soviets) prefer to be part of a country that is linked east-south-west and not locked up in an increasingly xenophobic and tyrannical Russia.
Most central asian people* have been made welcome in Kazakhstan, as it all helps to build population mass. I'm not there now, but I never heard of any Uighur issues.
* Funnily enough, "asian" Kazakhs are generally either chubby or skinny. The skinny ones are actual tribal kazakhs. The chubby ones are all in fact of North Korean extraction - shipped there by the soviets to staff all those catastrophic communal farming projects.