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Why do you think that? Yes, use a multimeter to measure between pins for shorts. Note that capacitors and parallel resistances may give changing results as caps charge up. Measuring Vcc to Gnd may give a reading of low K's of ohms as well.
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Maybe. If Vcc is shorted to Gnd, the PC supplying 5V may have a problem. If 3.3V is shorted to Gnd, that may fry the FT232 chip that can supply 50mA of current. If Rx, Tx or DTR are shorted to Gnd, that may or may not damage the FTDI outputs.
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Yes. Under Tools:Board, be sure to select the board type that matches your Lilypad.
You are not selecting a Programmer, the FTDI is just the USB/Serial interface, it is not a Programmer that connects to the ICSP header pins (SCK, MISO, MOSI, Reset, power, Gnd) to load a bootloader typically.
Why was the Mega so hard? All you have to do short its Reset pin to Gnd, connect up Power, Gnd, Rx, Tx, and press Reset on the Lilypad after the IDE shows "compiled xxx of 32xxx bytes". Or Press & Hold Reset, then release it when you see the message. Selecting File:Preferences and turning on Verbose outputs can make the timing easier to catch.