PSU - would recommend to change this (probably very old and deteriorated unit),
RAM - depends, if new motherboard supports your current type of ram. Is it DDR or DDR2 (most likely DDR)? There are LGA775 boards supporting either DDR or DDR2 also.
HDD - can keep,
Cooler - depends, if it has support for LGA775 socket. If it doesn't, then new cpu cooler is necessary.