Which is why the labour force surveys are considered the correct way to assess unemployment.

Unemployed normally means hasn't looked for work in n weeks, where n is a small integer, generally 1 or 2.

This is just about right. However, your definition implies that once a person stops getting unemployment benefits they are no longer considered unemployed. In general (International labor organization, ILO, definition) a person is considered unemployed if they worked less then a hour and looked for employment in the reference week regardless of whether or not their actually received benefits. They also have to be physically able to work in the 15 days following the reference week and only individuals 15 & older are included. Individuals who in the reference week stated they found employment but have not yet started their job are also included among the unemployed. In France there are two ways official unemployment statistics are calculated:

1. the ministry of employment calculates monthly unemployment rates based on the number of people who registered with the state unemployment agency (ANPE) as looking for work. This includes both individuals who received unemployment benefits & those who do not. Unlike the ILO definition these numbers count among the unemployed individuals who worked less then 78 hours in the last month unlike the 1 hour ILO definition. For more details here is a link in French.
   
2. The national statistics institute (INSEE) provides another measure of unemployment based on surveys of the population and using the International Labor Organization definition of unemployment. This is the number to use for international comparions. Right now it's 9.6% for the whole population for the end of November.

Good employment surveys also add questions to find out if people are underemployed & thus while they are working less then full time they would actually want to work more.

The one hour issue was debated a bit in France since apparently there are more people working temporary jobs (temp agency work etc..) then in other European countries but that's another issue.

Finally, regarding your question on what the "inactive" population is doing: they could be in school, have given up on looking for work at this point, do unpaid work (including childcare/housekeeping for their household etc..), they could be disables & thus unable to work. They real issue in terms of faulty unemployment statistics is the people who have given up looking for work and the underemployed.