You can apply for your daughters citizenship card and showing that you have applied for it, you can get her a temporary passport from the embassy. That should get her qualified for healthcare.
As for the husband, you have two options, outland or inland. You could start an outland application now if you show some proof that you are intending to move to Canada when he gets PR (like you being accepted to a university). You can also wait until you move to Canada to start it. It's not a requirement for him to be outside Canada to apply outland. You could apply through the US because he has legal status in the US or you could apply through his home country. Approving you as a sponsor will take around 7 weeks, see
http://www.cic.gc.ca/english/information/times/canada/process-in.asp#sponsorship but then the application gets sent to the processing visa offices, US or UK and will take some more time there, see
http://www.cic.gc.ca/english/information/times/international/05-fc-spouses.asp At the end of that process, he would have permanent residency. There is no open work permit while you wait for outland and if he's called for an interview, he'd have to go there for it (US or UK)
If you apply inland like you are talking about, you can apply for an open work permit and visit visa extension at the same time you apply for the sponsorship but getting the open work permit, even in a best case scenario will take 5-6 months. If you applied outland, he could have his permanent residency by then. With inland, getting the PR will take 1-2 years even if everything is ok and he's not advised to leave Canada during this process because if for any reason they don't let him back in, the application is gone. Another big downside with inland is that if they require an interview, they will not give him the first stage approval and work permit in 5-6 months, instead they will forward the file to your local CIC office to do the interview. Some of these offices are so busy that it can add 1-3 years to your processing time during which time he'd be sitting in Canada without a work permit.
As for officially moving to Canada on a visitor status, he's not really allowed to do that. Doesn't matter if he's applied for PR already or not. Officially, he is only "visiting" until he gets his PR.