I noticed that there was some new trunk-looking stuff that's coming around from these sides at the bottom to the middle on the thick stems but not on the top thin branches. What are these, are they new shoots, leaves or branches or something else?
Yes, these are new branches/trunks. Currants are usually multitrunked large shrubs or small trees. Your plant's trunks are in danger. See all that moss on the bark? Soil, rocks, gravel, moss on the bark of woody perennials will cause moisture on that bark so long bacteria can start using your plant to feed and water from. The bacteria will girdle, kill the vascular system that supports your entire plant. Pull the moss away from the bark and make sure the plant is not planted too deeply. Just the roots should be under the soil, never the bark or upper plant! I love currants!!
I would transplant and make sure the top of the soil is only 1" from the rim. Hopefully this pot has good drainage holes and no rock or gravel below the POTTING soil in that pot. Better would be to plant this guy in your yard in the spring AFTER hardening off from indoors? If out of doors should be fine to plant in your garden. Tough plant loved by birds, butterflies!! Beautiful plant!
Try NOT to use wood chips or bark for mulch.