2 week old baby allergies vs. cold also questions about umbilical stump?

ok my now 2 week old went to the doctor yesterday with a cough and sneezing. per my dr. she has a cold and i was told to treat her with a humidifier and saline nasal drops. as of this morning (after being up all night watching her) she awoke with her eyes basically glued shut, i work in ophthalmology and noted that it was white and white discharge is not harmful. i had to leave her with my mother-in-law and when i returned the discharge in now a little yellow. i am unsure if she truly has a cold or maybe is allergic to something in my house. i would imagine if she was allergic she would have shown signs of this a week ago when i first brought her home. i am unable to contact the dr until tomorrow to ask her this and im just wondering what anyone thinks. also, she lost her umbilical stump today when i was gone. i noticed before i left that it was a little irritated because she was grabbing at it. when i changed her i noticed that she was still a little irritated and that there were a few drops of dried blood on her onesie. is this normal? should i still clean it as the dr. directed?