the US and even India are officially secular countries, but separation of church and state does not mean reduction of personal freedom for practicing their own religion
Secularism can come in different hues. Often people make difference between Anglo-Saxon and French secularisms. China will retain the right to interpret its own secularism for sure. But, as far as I can see it, Chinese secularism does not want religion to be too visible in public places (government offices, schools, etc). Up until recently, Turkey, too, practiced this version of secularism although currently they are moving toward a Salafi-style state-sunni sect mixture. Again, China cannot take lecture from others' practices; if it works for them, well, praise to them.
both of the above states do not interfere with personal space of people to practice their religion as much as China
In the US, any religious (Islamic) activity carried out in the public is closely being monitored. Especially after the 9/11. It is mostly the minority behavior that defines the majority reaction. I guess it is sort of a cycle; violence breeds tougher measures, and tougher measures lead some into more desperate actions. But, here, it is not the state, with its absolute right to legal violence, to give in. The Chinese state cannot show weakness.
China as a one party much more authoritarian system, is expected not to give as much freedom, but people will compare and make this an issue, they will inevitably say that look religious people have so much freedom in the West or even in much poorer country like India (even though large number of people get killed in communal Hindu-Muslim riots every few years), how come China do not provide the same rights?
People have absolute right to practice religion in their private life. But, I think in China, public space is restrictied and it applies to all sort of religions and sects. The West, also, is not a unified space. For example, France has a tougher dress code in public area. Hence, it is not about China being authoritarian; every government is authoritarian by nature. It is about the rule of law that applies to everybody.
It is not like giving these rights will enhance their feeling for separatism some how, the effect I think will be exactly the opposite.
It depends, I think, on what sort of rights are being given. Public space rules can be relaxed (although I am against a gross display of religiousity in public) but it will hardly mean in China that they will be granted religious autonomy to run their own schools, to have their own separate civil law etc. The root of Xinjiang terrorism is that the terrorists do not simply want the right to cover their head or go to the mosque to pray (these rights are inherent to them already). They want rights for a seperate religious entity, such as running madrasas. This, in my opinion, will lead to more separatism and terrorism.
Also, let's keep in mind that, the few hard-core terror groups will not be impressed by any move by China as what they want is a separate political entity of their own. You may compare this with ISIS.
The restriction on some religious practices, is this only limited to Uyghurs or does it apply to all Chinese religious minorities?
It applies to everybody. Hence the demolition of churches without licence. In China religion is regulated by the state and state-sanctioned religious personalities. It may not be left to the wish of each religious groups. (Still in Turkey, all friday sermons are unified and state-sanctioned. You may not just go up there and speak your mind. On many instances, through friday sermons, the government propagates itself, legitimizes its policies. China will certainly not become like this; but, people, sooner or later, have to learn themselves how to be moderate, modern, and secular-minded while observing their faith in their personal lives and carrying out acts of goodness like community work and charity).