1) Yes, you can apply for PR for your spouse while you, the Canadian, are living outside of Canada. That is how our application started. We applied in February and I lived in Texas but I knew I was moving back to Canada in the summer (which I did). If you are living outside of Canada you have to provide proof that you intend to live in Canada when the application is approved. We did this by sending examples of where we were talking to real estate agents about rental properties we're going to buy for income, letter from my parents that we could live with them until we found our own home, proof that my house in Texas had been put on the market to be sold, etc. I also provided them with information about why I couldn't move back to Canada any sooner than the summer and a timeline of when I would be back to Canada.
2) I believe you can only get the work permit for your husband if he applies as an "inland" applicant which means he's already in Canada, but you may want to check on that. We applied for my husband as an "outland" applicant, because he's living in the UK while we wait, and he can't work until he gets approved for PR and immigrates.
3) I *think* it's either the biometrics or medical (or both) that have to be done outside Canada. You would need to check around on that some more to confirm. The medical is likely going to cost a lot regardless (it was £330 for my husband to have it done in the UK). It isn't a part of any health care services and you can only go to a physician that is on their list. I doubt it would be covered by any insurance in the US either.
4) I agree with
@patvicto that it could complicate things if you are applying to two countries for residence. At the very least, it could cause delays which will just extend the time one or the other of you isn't in your intended country.
I would recommend if you are looking at possibly applying in both countries, that you speak to an immigration lawyer for each country and get their advice.