I use my LeBass with in-ears, which should be the most effective test of how noisy the DI is. I've never noticed any noise at all. It sounds great to me, nice detail.
I would probably not leave the pedal on when not in use. Preamp tubes generally go a really long time before you need to replace them, but, as you've noted, the pedal is keeping the tube warm the whole time it's plugged in, so probably not advised.
I'm using a Shuttle 9.0 and a Hathor 1203, love the AudioKinesis stuff. Getting another 1203 eventually. My friend (Shrigg on TB) works with Duke as a beta tester and happens to live in the same town that I do, so he's responsible for setting me on the AK path.