Thanks everyone for the warm welcome.
I've ruminated on my first experience, and after doing considerable reading here I believe I understand why things didn't work out as expected. The two trees I used were about 20' apart, for one, and I suspended the hammock without a 30-degree angle between the tree and hammock (it was too straight). Because there wasn't enough downforce on the tree straps, they were able to slide down. My lesson here is that the further the trees are apart, the higher the tree straps need to be to get the proper angles.
Also, rather than using tie-downs and tying somewhere onto the strap, I'm far better off using a tie-down that is positioned so that the hammock is secured to the end of the tie-down strap and serves to hold it tight.
Finally, I'm considering how I want to position the tarp... whether to keep it the way it is or have a separate line.
Regardless, I'm intrigued with the idea of a hammock as shelter instead of a tent, and plan to play around some more once I return home.