Is it possible to custom program an AVR chip by putting it in the AVR socket of an Arduino?
Yes. You can program a virgin AVR by connecting it up to some GPIOs and reflashing it. If you had a second Arduino, the chip being programmed would sit in the socket.
(With the Arduino Uno, it's more complex it seems)
If you have an Arduino with a FT232RL chip onbaord (Arduino Duemilanove or earlier models or their clones) then you can bitbang the FT232RL to program the bootloader or any other code into the AVR.