Oh well done, Chick! That's great news! You can dab a little iodine on the navel, but if it's open I like to use some anti-bacterial, or antibiotic cream (one without painkiller in it), to hep prevent infection, and keep the little one on a sterile surface until it's sealed up.
That, I call it an "umbilical cord", is normal. It should detach by itself soon. It's behaviour sounds normal for a newly hatched chick. I would have a look at the navel and the surrounding area once the cord detached.
O.K. The first one, where is the chick's beak? I cannot tell much from that pic, sorry! Can you dampen the inner membrane and show/tell me what you see? Try and keep that membrane as damp as possible, just be careful around the chick's beak.
The second chick. What does it's navel look like...
The membrane should be transparent and any blood vessels clearly visible. If I'm unsure I remove a bit more of the shell and look at a "fresh" part of the membrane. Try on a different part and if you can take a pic so we can see?