Health insurance selection for a new job contract - Germany

I'm close to sing a new job contract with a German public service organization.
I'm italian and I'll be in German for work only keeping my primary residence and family in Italy where I'll be back during the weekend.

Since I will be above the monthly threshold of 4080 euro/monthly shown as a switch condition I would like to know if I'm free to select the desired one.

Reading some posts it seems that also if the previous work was outside Germany you need to demonstrate that the overall incomes of the past three years were above the German threshold, is that true? This is not my case.

Having a good "German" health insurance (public or private) is that means that also my family (wife and children) living in Italy could have a "fees refund" if they are using private doctor or dental cares in Italy (in Italy we have a public health assistance and you do not have to pay for public doctors and "standard" medicine, neither for hospital)?
If I go to a private doctor in Italy could I request the "fee" refund to my German insurance?

on http://www.prowista.com, there are several German insurances in comparison. maybe they can help you....
good luck!

If you can not show proof that your gross salary (and not any other form of income, btw) exceeded the legal thresholds during the last three year (basically if you start your job now Dec. 1st or jan 1st if would concern the years 2009,2008 and 2007) then you can only get yourself into public health inusrance IF you are an employee. with some luck, however, the new governement is soon going to abandon this stupid three year rule again, but for now this is your legal situation...

With regards to your family: they can only be co-insured in public health insurance if they are residents in Germany. If they are and reamin residents of Italy, they can not be covered from a German health insurance AFAIK

Cheerio

Just noticed this, too: only if you are in sudden need of medical treatment while in Italy, but not if the visit was planned by you or if the treatment could just as well have been obtained in Germany.

Cheerio