Yes, taking a shoot that formed away from the main mass is ok. They do not root very deep, so the branched off bines are ok. I usually just sever the runners with a spade. Yes, the original rhizome will grow laterally. Do not worry about damaging the original rhizome if you are 8 inches away from it.
You can also root clippings. I cut the clippings after 2 leaf sets and the bottom is below the third set, which I remove the leaves. That node will seal the clipping and roots will form above it. You don't want too many leaves on each clipping. You can use a bucket with 4 inches of water or a beer bottle. They take about 2 weeks to start rooting.
You can fertilize any time, but make sure that it does not have any weed killer in it. Frequent feedings are better than a dump.