Right, I think you're freaking out a bit here and need to calm down. Employer might not have answered due to being ill, friend dropping in, roof leaking etc - you can't be sure it's because she's pissed off at you.
You're in the worst bit of this now as DD just started preschool. Kids do get ill but the start of preschool and school is the worst. Hold your nerve and get through this bit and the illness will be less frequent. Do what you can to give her good nutrition, vitamins etc to help her fight off illness and get better faster.
I get what you mean about DH job, why rock two boats when you could just rock one? This is one of the factors that holds women back in the workplace tbh, as they're seen as less reliable. Although in theory you should take equal leave, in practice if you know it could impact your DH's progression and job stability, it might not work that way. He should really take at least some of the hit if not 50%.
If you think your employer is annoyed at you taking 100% of the sick child cover, you could always tell a white lie and say DH is doing some/did some before or after the leave you need to take.
For what it's worth, my parents are fairly close by and supportive and have DC quite a lot - but wouldn't usually step in when they're ill. This is because the kids usually want me or DH, and I don't think it's fair to give DP a bout of vomiting or whatever to thank them for their help. They sometimes help with a very slightly ill and non-contagious kid but it's rare.