1 out R3 on third, B1 strikes out swinging at a ball that bounces in the dirt & is caught by F2? Can B1 try to anvance to first?
Can he try to advace if the catcher drops the ball that is in the dirt, or that is in the strike zone that he just drops?

B/R Can Advance
 

B/R can advance to 1B on any third strike that is not "caught" (see definition of catch in rule book) when 1B is not occupied or with two outs.

1 - yes this is an uncaught strike.
A CATCH is the act of a fielder in getting secure possession in his hand or glove of a ball in flight and firmly holding it; providing he does not use his cap, protector, pocket or any other part of his uniform in getting possession.

2 - yes

See 6.09
The batter becomes a runner when_ (a) He hits a fair ball; (b) The third strike called by the umpire is not caught, providing (1) first base is unoccupied, or (2) first base is occupied with two out

Bob