Do you need to turn the warmer/steamer off or can you leave it at the desired temp consistently?
The temperature stops when the set temp is reached.

What is the widest size bottle you can put in this?
Two baby bottles of this size (3in X 8.46in) will fit in the bottle warmer.

Can you warm refrigerated breast milk and also water for formula?

Can the product/ warmer be used as a cooker? 
Yes. It has a stainless steel liner and you can cook noodles, warm yogurt or heat food with it.

What is the purpose of the white tray?
To raise bottles when warming and sterilizing.
You can also use it when boiling eggs.