As far as I remember, most nurses jobs do not qualify for H-1/E-3 type visas because a US Bachelor's degree is not required for the jobs. If you can find a job where a Bach. degree is the minimum requirement, you can be eligible for E3 or H-1B.
E-3 visas are available only for those jobs that demonstrably require a bachelor's degree in the field of work you will be performing. We recently provided assistance in an E-3 visa for a Shakespearean actor and teacher who was coming to teach acting classes in USA. He would probably not have qualified as an artist under E-3 visa, because to be an actor, a bachelor's degree in performing arts is usually not required. But as a teacher, a bachelor's degree was indeed required.
The employer should withdraw your application.
You do need a job. The company does not have to be a specific size, but it should be large enough to require a professional accountant (not just a book keeper).
Just go to Canada or Mexico and get your H-4 stamping and come back. That should work.
To the best of my knowledge - no. You need an employer.
What kind of problems can employment-based nonimmigrants (H-1, L-1, E-1, E-2, E-3, TN) face during reentry?
I think this question is most relevant for H-1 holders, but other employment-based nonimmigrants may also note the general principles here.
Applying for a green card should not be a problem for you or your wife. I always suggest filing two independent green cards where the option is available. This provides for a safety net if the process fails for one of the spouses.