Hi Stephani,
If it were me, I'd be charging anywhere between 2-3 grand. I charged $75 a day for all-day/overnight care as a nanny nearly 10 years ago. (I had a family whose parents liked to take longer trips sans children.) And that was a bargain, even then.
And as another poster said,I'd have her make other arrangements for the dog.
For what it's worth, this is not a job I personally would take. And from your explanation of the job, it doesn't really sound like you want to do this either. It would be worth telling this other mother to find a different option, now, while she's got time. I just read so many red flags in your post that I don't know how it's going to work for 30 days. Are you going to be okay if they are deployed for longer, because we all know that when it comes to the armed forces, their agenda comes first, families second. (I say this as former Navy: still remember in Boot Camp our company commander explaining "If the Navy had wanted you to have a husband and a baby, they would have issued you one.")
I hope for peace for you, whatever you choose.
Take care~
H.