Question one: Using WWII as an example. When nations are at war with each other how did they communicate diplomatically? To my understanding there were usually no embassies.
Question two: Was there much communication? What were some instances?
Then, as now, the opposing forces generally communicated through the embassy of a 'protecting power' - meaning that they had taken on responsibility for protecting their interests within a hostile nation.
In the early days of World War II (circa-1940) for example, the US served as Britain, Belgium and France's protecting power in Nazi Germany. Later in the war, Japan awarded the same status to Spain, Sweden and Switzerland in order to communicate with and ensure its interests were represented in the US.
In fact, Sweden and Switzerland, thanks to their long history of neutrality and physical proximity to the Axis became the protecting powers of choice with the former representing 28 governments and the latter representing 35 by 1943.
The Swiss even set up a special office in Bern with 150 staff (and another 2,000 employees stationed elsewhere) just to process this huge volume of diplomatic traffic and extra responsibilities, such as inspecting internment camps, issuing passports and exchanging prisoners (this particular responsibility was enshrined in the Geneva Convention) .
To give you an idea of the scale involved, Switzerland represented 219 individual embassies and consulates, trafficking 500,000 reports and 54,000 telegrams, and oversaw the exchange of 35,000 civilian prisoners.
Source: Britain, Switzerland, and the Second World War by Neville Wylie