Monks and Barbarians take 30% less damage than the other 3 classes. This is an inherent attribute valid from level 1.
Now, 30% is significant but I'm not sure it's enough to justify the radical difference you're describing. How certain are you that it's not just your general perception? Because in practice, a Monk does not tend to just stand there tanking. He tends to hit back, which often causes a "hit" animation on monsters which prevents them from attacking. This is before, of course, we consider that a typical Monk is loaded to the brim with skills that stun enemies, blind enemies, decrease enemy armor, increase the Monk's armor, etc.
All of this answer is for same-screen co-op. Online play is different. I have leveled two characters in coop from 1-8 with simultaneous level ups. Nothing special was done except that +xp gear was kept equal between them. I've also observed 3-player co-op with similar results.
Is exp divided evenly each time an enemy is killed?
Yes. If the players are equal level it's definitely even. There may be a little catch-up distribution in the case of non-equal levels.
If not, does the player who dealt the killing blow gain more experience? Is experience gain in any way related to the amount of damage dealt? Do I get exp for every monster killed by the party or only for the one I hit at least once?
No, No, and every monster killed by the party. Bonus rewards (such as Massacre from a kill chain) and Quests, also shared by the party.
Do I still get exp while dead (and waiting to be revived)?
I don't know this one - hopefully it doesn't come up much.
Do I get experience from special finds (i.e. books, scrolls, etc.) or does it only go to the character who activated the item?
Shared by the party. I have seen two characters level up at the same time when one picks up a book.
Is quest exp shared evenly among party members, or is there a way to get more exp than the others from a quest?
Shared evenly. Although it may be level adjusted.
Do items that increase exp gain only affect myself or also the other party members?
This is it, this is the only thing in the list that is personal. If you wear more +xp gear than your buddies, you will get more xp.
Money and loot for starters are individual.
Item loot is individual. Money is shared by copying. If one player picks up 10 gold, every player will glow and get 10 gold.
Best Answer
Everyone within a certain radius of a kill gets XP, so it doesn't matter whether you're 'putting in the work' or not. How much XP everyone gets is dependent on a number of factors, but the most likely culprit is gear. Have a look at your equipment and see if anything is giving you 'bonus XP on kill'(which is not necessarily 'bonus XP on your kill'!).
As @Timelord64 notes, Diablo level curve is exponential: each level requires more XP than the one before it. So if you're significantly lower levelled than your partner, you may find that you gain levels more quickly. Generally the level curve means that the level difference will shrink if players are earning XP at the same rate.