There is no weekly pass, but you can get individual country passes that will cover x number of days within 1 month, and select pass that will cover x days/1 month/3-5 conjoined countries. 
 
 

Railpasses cost 20% more if you buy them in Europe and not all passes are available in all countries.  As a non-European you generally can’t buy a single country railpass while you’re in that specific country — you probably can’t buy a German pass in Germany. 

There are specials, discount cards, “happy weekend” tickets, etc. available, but it’s hard to know what will work for you without a combination of luck (finding a ticket agent that’s super helpful and willing to sell you a ticket that may technically be for residents), language skills and knowledge of the local rail system.

Your best bet may be to buy a couple of flexipasses or select passes.  Use a pass for any long, expensive trip (ones that cost more than a day of the railpass).  Buy a point to point ticket for any shorter, cheaper trip. 

 
You can, but there is a special ticket counter for it, kind of a tourist only section.  And as you said, it costs about 20% more and you can’t be a european citizen.