So...
There's this NPR. I havaen't been able to get him to communicate with ME at all, and if I go to any of his systems, he tells me to leave.
A request to leave can be ignored; your diplomat will probably be able to raise their opinion of you faster than the penalty for being asked to leave, especially if the only ship they can detect is a properly–designed diplomatic ship.
However, there's this one ship in a remote system that I was able to park a Diplo ship next to.
But of course this is even better, since your diplomat can work unimpeded.
I was just informed that they now grant me trade access... despite, technically, my relationship with them being 0.
As Andrew already said, that is only your people’s opinion of them. It only controls what treaties you are allowed to offer them. Your diplomat has already raised their people’s opinion of yours enough that they have offered you a trade treaty. If you keep raising it, they will give you more stuff.
What I want to know is, what do I do now? If my civvie lines decide to send freighters to their space, are they going to get attacked?
No, because they gave you a trade treaty. In fact, each successful delivery by one of your ships to one of their colonies will raise their opinion of you. This stacks with what your diplomat can accomplish. Of course for that to happen you need to have a continuous path of stabilized jump points to their colonies and back, and the trip cannot be so long that your civilian freighters never choose to make that trip. You may need to found a new colony specifically for the purpose; just don’t put it in a system that they already claim.
Or should I just wait until I get so far as they think of me as an ally, then just reposition my diplo ship in one of their home systems and wait 40 years for them to "catch up" in relationship rating?
That will probably tank their opinion of you; the penalty for leaving a ship in their home system is over 20× worse than in a minor outlying system that they have claimed. A properly constructed diplomatic ship garners the least possible penalty, but it is still pretty bad if you camp out in their capital system. As their opinion of you goes down they’ll withdraw any treaties that they have previously agreed to, and if it gets bad enough they’ll go hostile.
In order for your people’s opinion of them to go up, they need to build a diplomatic ship and send it to your territory. Sometimes the AI manages this, other times it doesn’t, or perhaps it decides not to. (Naturally, the two are indistinguishable to the player.)