You can go to a third country, but your entry back into USA can be declined if CBP believes you are spending more time in USA than you should. Typically, B visas are for brief visits (may be six months or less in a 12-24 month period).
Tourist visa seems to be appropriate. Times vary by consulate and are usually very short -- often visa is issued the same day an application is made. You can also get an appointment for emergent reasons.
I believe you can take training on B-2 as long as your main purpose of stay in USA is is still medical treatment. Getting paid may be a bad idea.
A B-1 visa is appropriate for medical treatment.
You will need a consultation with a lawyer. It appears you were formally denied admission and excluded from entering USA. The one-month detention sounds strange. Anyhow, get a lawyer, this is not a simple matter.
Your chances are slim, but it is impossible to predict.
She needs to contact the US consulate closest to where she lives. Normally, they will be able to tell her what she needs to bring and what the procedure is. This information should also be available on consular web site. Most consulates also respond to e-mail enquiries.
I see no reason why not.
In my view, a B-1 would be appropriate because you are not engaging in any employment that would take a job away from a US worker and you are not getting paid for your appearances.