Diablo – Do you still get the vampiric effect on illusions

For attacks that hit multiple enemies, Yes.

I haven't explicitly tested all on-hit effects. I will update this answer as I do.

Life on hit: Confirmed. With a +2 life on hit weapon equipped, I often saw healing popups of +6, +8, +10, etc while using area attacks that hit multiple targets.

Chance on hit to fear: Confirmed. Using Deadly Reach I saw two enemies get feared and run away.

When dual-wielding, you hit alternating with your main and your off-hand weapon, receiving a 15% attack speed bonus. So in the end, you'll get half of that damage bonus if if you only have it on one of your weapons, but the full bonus if you have it on both or on any other accessory. But the end effect is that you get the average, as you also only hit with the average of your two weapons.

So you should in general only dual-wield with weapons that are very similar in damage, or you'll do less damage than you could with a sword and shield combination.