It's because Southern California is populated and affluent enough to support two NBA teams. The TV market there is huge.
No. The Clippers have made the playoffs four times, losing three times in the first round (Utah twice and Houston). The only playoff series they won was in 2006 when they defeated Denver in the first round. They lost to Phoenix in the second round.
The LA Lakers and the LA Clippers both play at the Staples Centre. They are able to do this because the floors that the Clippers and Lakers play on is a removable and moveable floor that disassembles just like blocks of lego. Therefore when there is a Lakers game and the Clippers floor is out, then the staff at the Staples Center just switch them out and now its the Lakers home court.