I am considering moving from southern Alaska to Central or Eastern Europe
My comments reflect all possible country options in Central Europe (which is the geographical basis):
https://en.wikipedia.org/wiki/Central_Europe
I would say:
Depends on the country, where you live in the country, and how you want to live. Cities are more expensive than in the countryside.
Also, Europe is all over high density. Upon first moving to Europe from the US West Coast, that was my biggest "culture shock" : cities, towns, and people were everywhere (development similar to US East Coast). Of central Europe, Poland has the most undeveloped land where you might be able to "get away from it all".
How does the cost of living change outside of the big cities?
It is cheaper to live in the countryside versus a city. Transportation options are also better in Europe than many parts of the US. For example, you can live along the main rail corridor between Vienna and Budapest and get to either city very easily in very nice, comfortable intercity trains. Houses and properties in the country side are much cheaper in Hungary than in Austria.
How hard would it be to get the visas and permits necessary to do this move?
Depends on the country. You have to contact the US based Consulate of any country of which you wish to become a resident, and ask about their residency permit application process.
What countries have good universities/colleges for my son? He is particularly interested in Anthropology or Psychology.
For Psychology: Austria (Vienna). After all, it all started there.
I am also interested in the possibility of owning a business once we are established in the country. Which countries would be good for doing business in?
Switzerland (yep, from the above link it is actually part of Central Europe).
Good business needs legal consistency, which as already noted by szocske is lacking in places like Hungary. Switzerland has a solid and dependable legal framework, low VAT rates, low income tax rates, etc. Setting up a sole-proprietor business in Switzerland is drop dead easy if you have a residence permit, but you can also setup a "limited" company for 20,000 Swiss Francs.
However, living in Switzerland on just 2500 USD a month would be difficult.
Many small businesses are currently going bankrupt here in Hungary. The economy is depressed.
Most (if not all) countries in Central Europe (including Switzerland) have a pre-pay income tax system (not pay-as-you-go like in the US), so be prepared to fork over hundreds, if not thousands, of dollars a month in taxes, even if you have no business income.
Also : The US has no Social Security tax agreement with Hungary. Any business income you make in Hungary may also require SS tax payments to the US Social Security Administration.