Yes, we would. 3-5 years is a long time to go without food. Consider that American cities are only stocked with 72 hours worth of food. The biggest/baddest (heavily armed) would be the only survivors. Most large animals would die too.
During the Toba volcano 75,000 years ago, only 2,000 humans world wide survived...that event can still be seen in the genetic record.
Depends on who you mean by "we." Western civilization would be damaged beyond repair, as we know it now, but it would certainly survive in some context. Military bases in isolated regions, nuclear subs and other such well stocked and well coordinated entitys would maintain their technological sophistication.
Think of a well provisioned nuclear aircraft carrier, with machine shops, etc., and assume that it survives the initial impact. Joe suburb might be in big trouble, but after he runs around for a few weeks in rags playing survivor imagine what happens when the USS Ronald Reagan pulls into port and declares martial law?
Now, the poor schmucks in Rio or Delhi or Athens are screwed. Ironically the folks in the hills outside Karachi or in the hinterlands of Nepal, assuming their terrain isn't imolated in the blast zone, should be fairly well off, as they're used to making do and "roughing it" in "primative" conditions.