I have worked through colds, migraines, zero (and I mean ZERO, due to a hospital visit that went all night) sleep, pregnancy woes, and a number of other illnesses. The beauty of our job is we can stay in our jammies and sit on the couch while the children play, if required. And maybe catch a nap at quiet time, if we're lucky.
That said, I have had to close a couple times. Once because our entire family, except my hubby, came down with the worst stomach flu I've ever experienced. We didn't have enough bathrooms to accomodate all the sickness!! Ugh... And once because I was ok, but my kids were incredibly sick.
I will call parents as soon as possible. I open at 7:30, so I will call people at 6am. It gives them and hour and a half to find someone. If I am outright cancellling on them, I will chalk it up as one of my paid personal/vacation days. If I call and give them the option of coming, and they choose to stay home, I still get paid. So far, all the parents I have dealt with have been GREAT. But I make it very clear, upfront, that I have 4 young kids and that on occasion, chances are I will have to shut down.
In regards to the hours, I know that some people work out extended care fees, etc. I don't. My hours are my hours, and I learned early on that I have set them that way for a reason. Is the child going to be picked up earlier, since she is getting dropped off so early? I would hope so. The only caution I would give you is; if you say yes, don't get resentful of it later. I wouldn't want you to take this family, because you are really in need, and end up hating that you said yes to a 6:30 drop off. Then you'll need to change it, and she'll get mad, because you did agree to it! And another thing, if you told her your hours and she ended up asking for different hours anyway, I always proceed with caution. This type of mom may feel like she is in charge, and in the future, attempt to dictate other rules, etc. Of course, she could be totally nice, and just liked you and was hoping you could accommodate her. I am just speaking from experience.