1) Saying the D/G strings are naturally muted wouldn't be accurate, not would it be to say that they're going to inherently produce noise (that we can detect). Unless you're sloppy with your fretting or your fingering, you shouldn't get much vibrations from other strings unless you're using them, then moving away. (From what I understand, a double bass is a different story but I can't speak from experience there.) Listen carefully, watch yourself - if you hear other strings sounding that shouldn't be, you're probably touching them somehow and sometimes it doesn't take much: practice slowly--to acquire accuracy--if that's the case.
2) It depends on you and your accuracy. If you're hitting the A (or others) on accident and you don't want that, you'll need to mute whatever needs to be muted. If you're on the E and you're hearing the A, one/some of your fretting fingers resting lightly on the other strings can help; this should be physically effortless, mostly: no need to apply much pressure, just enough to keep the string from vibrating if you accidentally hit it while fingering.
Don't get too caught up with this. Practice slowly and pay attention to what's going on with your hands and the sounds you want/don't want - you'll get it.
On a side note, the first riff I ever learned was the intro to Metallica's My Friend of Misery. I didn't find it difficult but it was a cool--for me--introduction into accuracy across strings and using three fingers (I don't use a pick much). This may be a horrible example--or bad advice?--especially considering there are some great tutorials out there. But you might try it for having a little fun while practicing across strings (not much muting going on as discussed above, however).
TalkBass: where some of the simplest notions turn into rocket science.
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