So I've slowly been figuring out how to make other empires like me but I can't figure out how to get my own rating of them up to even positive levels. It seems like I'm at the mercy of the AI doing my missions and sending envoys. I have the other empire's rating of me at around 10 but that doesn't let request any agreements or pacts.
This is actually not a problem for me at all, even when playing versus the tougher AI types. At the start of a game, when you're shooting for the magic 4.50 relationship number for a Ceasefire; just offer them missions they can complete or reject quickly. That is, offer them the "Give Credits / Metal / Crystal"; it takes the AI about 10 seconds to complete or reject them, and you can keep offering them over and over again to boost your empire's perception of their empire.
It's also a fantastic way to 'supplement' your income, to the point that you can very quickly pay off the initial costs of getting the Diplomacy tech to offer missions in the first place! I use this so much that my empire's feelings to the AI reaches close to the perfect 20 before I even offer the most basic of pacts.
In the beginning ask for 200 units of metal or crystal. I find the AI tends to complete those more frequently than the 1000 credits version. For sure, ask for the resource type your economy needs the most, but if you have an idea of what type of planets the AI holds, you can improve your chances of success as well (AI going for volcano planets? Ask for metal!).
Also, the AI won't honor your resource requests until their relationship with you is about 1.0 to 1.5. If you find the enemy rejecting your requests (along the lines of "It's laughable to think I'd help you!" as opposed to the "Eh, we can't do this right now"), send them resources or dance an envoy ship around their gravity wells until it comes up high enough. Relationship gained from resources can only go up to a cap of +5.0 with the right research, so you won't be able to quickly sway a sworn enemy.