Does an ARM (LPC21xx) Programmer pulls P0.14/BSL to ground?
As starblue points out, you are referring to the LPC range of microcontrollers. Mentioning this in the title and/or text of your question is advisable.

Whether the programmer pulls BSL low for you (and also whether it resets the chip for you!) depends on the programmer and the PC software you use.

FlashMagic and lpc21isp can be configured to do both for you, provided of course that the programmer hardware (which can be as simple as an FT232RL chip) supports this and is connected to the target correctly.

I cannot make sense of your last question. Maybe you want to know whether the bootloader will spontaneously jump to the application after a certain timeout? AFAIK not.