Flying from Beijing to Istanbul through Amsterdam in mid January: if the flight is cancelled, what are my options?

Unfortunately this depends a lot on the details of your specific ticket. I'll answer it assuming you actually have a layover in the Netherlands so that EU 261 applies.

Do airlines usually issue a ticket refund in case of cancellations due to COVID?

In most countries they are legally required to give you full refund. However, many airlines will try to avoid this by offering you a voucher instead. Many of this vouchers are poor value: do NOT accept them. You may have to insist and threaten legal action to get your refund.

Do they also allow a reschedule of a flight in case of cancellation?

They often do, if it's available at low cost to them.

If so, what is the flexibility there? E.g. can they give us a flight of a different airline? Do we have a say about which flight we get?

In most cases you do NOT need to accept a reschedule. You don't really have a say, but you can certainly try to negotiate with the airline for better conditions.

Can we choose between the refund and the reschedule?

Typically yes.

Where can I read about such policies on our ticket? They don't really write too much on the booking confirmations and airlines websites...

That is tricky because the rules and laws that governs this is an intersection of the airline policies and local laws in the countries involved. Since your itinerary currently goes through the Netherlands, you are currently covered by regulation EU 261 (https://europa.eu/youreurope/citizens/travel/passenger-rights/air/index_en.htm) However if you get rerouted through a non EU country, this wouldn't apply any more, so you need to research all applicable laws in China, Turkey and any layover country.

Are we still getting compensated for the ticket in full if only one leg of the journey gets cancelled but not the other?

Generally yes. You can't fly the itinerary as booked, so it's invalid. Again, the airline may still try to talk you into accepting a voucher or rerouting.

Now I realise the answers may vary from airline to airline. What I am looking is a general rule-of-thumb advice from experienced travellers.

Unfortunately there is little "rule of thumb" here since there is an almost infinite combination of airline rules, specific ticket conditions, and local laws in the countries involved.