Those brown parts are remnants of the sheath that protects the new leaf while it is forming and emerging. Totally normal you can leave them there.

You'll want to give it a lot of water over the sink so it can flow out unabated. You'll also probably want to change to a glazed or plastic pot to hold in moisture. Clay pots such as the one you have this plant in are very porous and evaporate water very quickly, maybe faster than you'll want to spend time watering it.