Yes - cut the scraggly older stems right back to about an inch, preferably to just above a node, which will encourage new growth, either from the roots or from the stem you've cut back. You can use the healthy, leafy parts you cut off to make new plants by trimming them to about 4-5 inches, strip off lower leaves to leave 2 or 3 at the top and pop the stems ...
They're unlikely to be roots off any shrub,hedge or tree nearby. They look horribly like bindweed roots http://home.bt.com/lifestyle/house-home/gardening/bindweed-what-is-it-11364193020893 but that can be confirmed or denied when a leaf or two appears. I take it you don't recall anything growing there prior to digging and covering it for winter?
This is peperomia obtusifolia, AKA baby rubber plant.
They can grow in water too and need to have only few nodes in water.
They like moderate to bright indirect light and are very resilient to pests. Soil needs to dry out completely before watering.
Image from Plantsrescew: