How to test-ride a full-suspension bike? I am looking for a MTB, but am unsure how to measure if the frame would fit me.

I haven't had a full-suspension bike. Furthermore, I am pretty sure MTBs perform entirely differently in the city and when going down a rocky track. My previous bike was small, short and awkward on pavement, but incredibly stable on the descents.

When test-riding, I very much doubt that any shop will let me ride for a whole day. And from doing a couple of circles around the shop, rushing a couple of curbs, feathering or squeezing the brakes and doing a jump or two there is hardly any feel as to what riding will be like, when I get used to the bike.

Kind of like with new shoes - at the start, every model is "awkward", as one has not gotten used to it.

I will ask the guys at the shop to help me with fitting, of course, but concerning the test ride - how do I do it correctly? What information is most essential to gather during the very short ride and how do I do that?
Bike shops have a special test-bikes to be lent for a whole day or even a weekend. I tested Specialized, Scott and RB for the whole weekend this way. 

You will pay them a refundable deposit (20-80% of price of the bike - depends on the shop). Some of them will charge you a little for cleaning/service or whatever. This may vary across the countries; I have experiences only with mine, but Specialized and Scott are international brands, so I expect similar offers around. 

This is an answer for the topic, not your last question, but I think this is the right way to choose a bike.

There are also test-centres in some Single-tracks where you can borrow a bike for couple of rides. But they have a limited offer of models there. 
Or some shops organize test-events near nice places to ride with many bikes to try if you book in advance. 

This is the best way to choose a bike. I was in the same situation as you. On the first test I borrowed two bikes from one Shop (RB): similar hard-tail and full-suspension and rode it with my friend exchanging couple of times to discover if is it even worth to invest to full suspension. It was quite convincing. Then I tested couple of other models of brands above for weekends. It's a fun to have a different new bike on every weekend. From these tests I came with very certain conclusion what I like the most.