Can I access individual pixels on my 16x2 LCD display I have a generic 16x2 LCD display connected to my Arduino.  I can use the standard LCD library to display characters, but are there way to turn on and of individual pixels?
It's possible to define custom characters for “generic 16x2 LCD displays” [ie, displays based on HD44780 or equivalent controllers], but only up to eight such characters at the same time.  See LiquidCrystal.CreateChar for details.  See page 2 and part of page 1 of the Print degree symbol ° on LCD arduino.cc thread for create-character examples that don't use the library.  Page 1 of that thread also shows a table of all the characters built into typical HD44780 controllers, some of which may perhaps meet your needs.

For pixel-by-pixel LCD control, look at graphics LCDs, eg the KS0108 or LCD12864 and similar.