I don't understand why there would be nastiness from you or him? They seem like they have their heads screwed on right, for the most part. The "Leaving her alone for an hour or two" is a bit of an overstep, but could just be because they don't understand how it would affect your DD and not as a "trying to steal her" kind of way, iykwim??
I think the best thing you can do atm is have an open and honest conversation with them both. Explain that you're happy to have meet ups with them for a few months and your DD, let her get used to them, and see how things go but before she stays over-night at their place that you'd like court orders or whatever in place, just so everything's upfront and clear and there's no confusion from anyone. Not that you don't trust them or anything, but simply so that it's all out, no problems or anything.
I think it's actually very reasonable to request(not to EXPECT, but to request and suggest) that you stay at their place with her, before she starts overnights alone. It shows that they are thinking about your DD, her comfort and well being and they understand how difficult the transition would be.
Honestly, as hard as it is, take a step back from the situation and look at it from all sides. Don't be defensive with how you feel about things, but try it with an open mind. You might find you can make decisions easier, and you won't be as paranoid.