The Kazakh language requirement has been ongoing for a few years now and it is extending to more and more areas, so that is no surprise at all.
The IIN is a new one, just get it, it relatively painless, although I'm confused when you say that you will only be there for 30 days. You must lodge your documents in person, when it is lodged, you will be given two dates, one will be a mandatory "cooling-off" period of 30 days date from lodgement when you must return to the office where you lodged the application, to confirm that you both intend to commit to the wedding, the second date will be the earliest date that the wedding registration can take place.
This is generally a few days after the mandatory "cooling-off" period.
As for the document regarding children, you need to make it absolutely clear with the Kazakh end precisely what they require (the name of the document and what they expect to appear on it). The former CCCP states often have arcane documents they require and these confuse the hell out of embassies worldwide.
Regarding the apostilles, as mentioned earlier, the UAE is not signatory to the Apostille Treaty (Hague Convention Abolishing the Requirement of Legalisation for Foreign Public Documents) and therefore you have to do two sets of legalisations ... rather than just the simple single apostille for the Certificate of No Impediment To Marriage as you would if you were from Australia or the UK (or any other of the 110 countries that are signatory to it).
So you will need to get two sets of legalisations rather than the single apostille for all UAE documents that would otherwise require apostilles.
You will also require a certified copy of immigration card (no additional stamp is needed despite what some notaries will try to tell you, ask them to call the immigration police and they will resolve the dispute, or just go to the next notary, there are notaries every 10 metres in Kazakhstan).