How to tell whether email seeking paid peer-reviewers for grant scheme is legitimate? How to respond?

This is most likely genuine, in particular if the proposal to be reviewed fall broadly into your area.

Just check the actual web address of the links given (rather than what they say they are) and/or search the respective institutions online. I did once a similar job for the Polish science foundation.

Actually, if you click the link http://www.ncste.kz/en provided, you will learn that they are attempting to boost their science standing and one way to do that is to get international reviewers.

Don't be too paranoid about viruses, just don't do stupid things (like typing your password into some unknown/new website), or opening mail attachments of unknown origin, etc..

I performed 8 reviews for them back in November of 2017. While the grants were quite low quality in terms of the science, everything seemed legit in terms of the process, paperwork and contracts. While payment was promised by December 2017 I have yet to be compensated, initially it was just a constant string of excuses and delays but now stopped answering my e-mails. I will update if I ever receive payment.

UPDATE: Payment received. While it was delayed by many months, I did end up receiving full payment for my efforts.

Well, the telltale sign of a scam is when payment time comes. There are basically four possibilities:

1. no payment: you have been scammed into doing a service gratis. However, the service you are doing here is unlikely to be useful for scammers.
2. payment "forthcoming" once you forward some processing fee/administrative fee that is small in comparison to the expected payment. Of course you never here anything once you pay. "small in comparison to the expected payment": the expected payment appears too small to make this scam worthwhile for this case.
3. You'll get the agreed fee.
4. You'll be paid quite more than you are supposed to receive, some flimsy excuse is given for that and are asked to move the excess payment back or onward to other reviewers. This is money laundering or fraud, depending on whether the owner of the original account is aware of the payment happening. In the former case, the police will eventually catch up with you. In the latter case, the money paid to your account will get retracted eventually and the money you "handed on" is gone irretrievably out of your own pockets.

So the real danger is case 4): if anything like that happens, notify the authorities and refuse to handle any money outside of your own pay.

Outside of that specific scenario, there does not seem much of a point in using this basic setup with the mentioned sums for fraud.