Any character can use any weapon; there are no limitations (except size - a medium creature can't use a storm giant's axe or a pixie's dagger). If they are proficient in that weapon they get their proficiency bonus; if not, they don't.
A wood elf cleric is proficient in longsword, shortsword, shortbow, and longbow (from wood elf) and all simple weapons (from cleric). Further, if they are of the Tempest or War domain they are proficient in all martial weapons as well.
By the way, your DM is really old (like me); the sharp weapon thing went out in 3rd edition - that is, last century.