I finally received the little Steinberg UR12 audio interface unit today.
I do get some background noise when I crank up the volume. But it's more of a normal background noise, such as you'll get even from a cranked up amp. I'm especially not plagued by the "Morse code from space" interference anymore! I can kind of hear some of it faintly. And it does seem that some PC noise travels to the card through the USB connection. But it is immensely more tolerable. I think I have to surrender to the fact that I'm in a noisy electrical environment and will always have to reckon with some level of interference. At least, taming it a good deal is a definite success.
As I expected from a couple reviews, the UR12 does clip very quickly when I plug the bass directly into it; or rather my preamp DI, either through 1/4" or XLR. I have to set the input volume below 50% lest it clips, and I can't really turn up the headphones volumes very high to compensate. (I haven't had a chance to test it out on the monitors yet.) Probably a good thing for my eardrums, but I really find it to be a bit too low for my taste when jamming. Recording levels seem fine: it's just the perceived volume in the headphones.
I've tried plugging only into the audio interface. That's when I find the volume to be way too low. So I've tried other configurations using my mixer (Mackie Mix8). Presently, I plug in like this:
Bass (+ pedal chain and preamp) -> Mixer
Mixer Aux Send -> UR12 1/4" line input
UR12 Monitor Out -> Mixer Tape In
Mixer -> headphones and speakers
With this configuration, it's decently quiet if I compare to the amount of noise I was getting. I especially get a good amount of control over volumes through the mixer. By keeping the mixer in the setup, I can also plug in other instruments, which I can then decide to send to the audio interface for recording via the mixer's Aux Send.
Opened to comments if you have better solutions.