What was St Mary's and Joseph's last name? and why are all people from
that period known by their first name.

Family, or surnames are a relatively recent practice.

http://en.wikipedia.org/wiki/Family_name

Yep, they originated, for example, in Ireland around 1100 A.D.  As a
result of this late naming, there are people with similar DNA but
totally different surnames since they often took different names,
though perhaps of common ancestry.

If there was some sort of family name in Biblical times, it was
probably a patronym, as is still common in Arabian and Semitic names: 
ben, bin, ibn =
"son of", as in bin Laden, or ben Gurion (an assumed name, son of the lion).

Good point

- there is also the trick of calling a father after his son

  Abu-Masen for example