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When dean was in nicu they cleaned it with alcohol... somewhere between that and my 3rd baby, they stopped cleaning the cords with alcohol, I accidently knocked maks off changing his diaper it was kinda wet looking, I just cleaned it with qtip and alcohol, by next day it was drying up and startd getting crusty

When my sons stump fell off it looked not healed all the way. At his 1 month appointment his pediatrician cauterized it chemically. No pain, and it just took another week to be fully healed after that. Ask his pediatrician to check it out.

Once the stump falls off the middle has to heal too. I just treated it as though the cord was still on until it healed too.

As long as it doesnt get smelly its fine. It needs some air to dry up  
I've never used alcohol. And I bathed them as normal and patted dry the belly button not rubbed.