1. When is the latest and earliest I can apply for the extension of my visa?
The latest you can apply is the expiry date of your visa, the earliest you can apply is 28 days before you reach 30 months in the UK.
You entered on 29th October 2013, so you will reach 30 months on 29th April 2016... which means the earliest you can apply is 1st April 2016 and the latest you can apply is 2nd July 2016.
2. Can I apply for visa well in advance? I was thinking of travelling to KZ in April and was wondering whether I can apply for visa now, in January basically.
No, as mentioned, the earliest you can apply is April 1st 2016.
You could either apply in person at the beginning of April (which costs an extra £400) for a same-day decision (you would get your BRP visa in the post within a few days), or you would have to wait until you got back from KZ to apply (as long as you can get it in before 2 July).
The visa fees will likely increase on April 6th 2016, so if you can get your application in between April 1st and 5th, you wouldn't have to pay the higher fees.
However, f you apply in January, your FLR(M)will expire in June 2018, but you won't be able to apply for ILR intil October 2018... which means you will have to apply for ANOTHER FLR(M) (at a cost of over £1,100) to bridge the gap between June and October 2018.
3. I understand that there were changes in visa rules. I previously took Cambridge KET exam and got 100% for it. Certificate says that it doesn't have expiry date. Can I use the result of this exam/this certificate again or should I take an exam which appears in the new list of approved exams (I didn't see KET there)?
I'm not sure I'm afraid, as I don't know how the exam certificates work... most people applying here on UK-Y are from the US and don't need to take any English language exams for their visas.