As I understand it, your diplomacy team will increase the diplomacy rating over time, dependent on their team skill. As your diplomacy rating passes certain milestones, the race will change their position towards you in regard to certain things, such as thinking of you as Friendly instead of Neutral, or giving you their Geo scans, or giving you research. You can't actually propose deals as you would in something like Civilization or Master of Orion.
As for why it's ticking down, even though you have a diplo team assigned, I guess if the team has a low skill, that might contribute. Or if you're attacking them (in my experience). Or maybe they're a spoiler race... I just dealt with what I thought was an NPR. Couldn't establish communication, diplo rating continued to tick downwards despite a team with skill of 140 or 150.
I haven't found a good way of finding their home base besides just exploring more systems. Even if they give you grav and geo surveys, that doesn't display their occupied systems.