Just watched part 3. Man, this game looks tough!
You asked a couple of questions about EUI during the episode that I don't see any answers for yet so,
Possible trades are indicated by the little square icons to the left of the leader. For strategics, they must want the item (at least one) and be able to pay the equivalent of 1GPT in either gold or GPT. If you look about 17:00 you will see the icon for iron for just about all the civs. The fact that Rome wouldn't pay 2 gpt despite apparently being friendly was a hint that he might be deceiving you.
For their luxuries to show up, they must have at least two copies (one may be imported) of the lux. For your luxuries, the icon will still show when you have a single copy. For your one to show, they must also be able to either swap for one of their own or pay the equivalent of 5 (I think) GPT in any combination of gold/gpt.
Essentially you can see possible deals, but you still have to decide if you want them and negotiate a price. The most useful part is seeing when luxes become available, and also who might buy your excess luxes.
Towards the end of the episode, one of the civs had two luxes he would sell. Rome also very briefly had copper to sell. There were also potential buyers for your marble, but you only have the one copy. Still, in various circumstances, you might want to trade for tactical purposes.
Regarding the info when you mouse over the leaders, generally red means "no", "bad" or "can't", while green is the opposite. The red embassy just means you don't have one. The next section of bullet points is diplo modifiers - green here is good. Finally there is a section on their relationships with CS and other civs.
Hope that helps.