(At least in the U.S. Do EU citizens have to report themselves for such transactions?)
http://ec.europa.eu/taxation_customs/individuals/cash-contro...
In Greece they have even extended this to other comodities such as gold and you have to report it regardless if you're moving it or not, otherwise they can freely confiscate it.
http://www.zerohedge.com/news/2015-12-01/greeks-told-declare...
No, but transactions over X € must be conducted using a traceable medium - cheques, credit cards, bank transfer, sometimes Bitcoin. X varies by country.
Conducting transactions above X € in cash doesn't automatically constitute criminal money laundering, of course, it just gets fined.
As for eliminating cash, yes we try to do that, but the reason given is not money laundering. The two reason/excuses which are commonly given are 1) preventing tax evasion and 2) preventing store robberies. Also note that the ones most heavily pushing for elimination of cash is not our government but our banks. I think card payments are much more profitable for banks compared to cash. But it is not like our government makes a secret about that its regulations surrounding cash payments are there to prevent tax fraud.
And the worst country for when it comes to hunting taxes is probably the US. A couple of days ago I had to sign two forms when opening a bank account where I guaranteed that I had no ties to America. Many smaller European banks wont even accept American citizens as customers due to the extra paper work to report stuff to the American IRS.