Approach when weather condition is below minimum
You haven't told us what country you are referring to.  I am going to talk about the United States regulations.

There are two separate regulations that come into play.  Part 91 and Part 121 or 135.  

Under Part 91.  The pilots are allowed to commence any approach (irrespective of weather) and determine if they can continue to land based on the criteria listed in §91.175(c).

Under Part 121 / 135.  The pilots can only commence an approach (proceed past the final approach fix) if the latest weather received is above landing minimums.  Mainly, pilots are concerned with visibility.  They will not be able to use "flight visibility" as the weather visibility is most likely different as the airplane descends towards the ground.  The pilots must use the weather reported from the airport.

To answer the second question, if the pilots are past the final approach fix and new weather arrives indicating the airport is below minimums, the pilots are allowed to continue the approach to minimums to see if they can land per §91.175.