Is there any way to see how much damage different type of mobs/NPCs do on my character; in particular, is there any way to see what kind of spell bbq'd me with how much damage inflicted?
Diablo – How to measure combat in Diablo3
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Related Solutions
Aside from practical experiments, you can also do some theoretical study using Blizzard's skill calculator here, and if you want to play it safe early on and choose only combinations of skills that are generally useful together, you should use "non-elective" mode for selecting skills.
The details are explained in the below video.
You pretty much nailed it. All your Intelligence, +Damage, Crit Chance, and other such stats are rolled up and multiplied out to give you the number pictured. All your ability damage, even as a Wizard, is based on that.
Here is a related question with a great answer containing details on where that number comes from.
In your example with Magic Missile dealing 110% weapon damage, if you are all-out casting Magic Missile, you can expect to do 110% of pictured 39.03 per second (which would be around 43 DPS with Magic Missile).
There could be other factors, like debuffs on enemies, that further change how much damage you do to those particular enemies. But the number pictured is your general overall damage value.
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Best Answer
There's option in the gameplay to display floating damage numbers on monsters and/or players. (by default you only see crits against monsters). However, these numbers are rounded to the nearest whole, while damage is certainly float-number based (if you watch health bar with level 1 enemies and a new char you can see this).
You could probably get a screencap system to create a damage log based on this, but that's far beyond my programming skill to code. Theoretically you could also intercept it from the outgoing server traffic, if you could figure it out. (assuming it's not encrypted, it should be pretty easy to do, but also way beyond my ability)
Also you do get a readout of what killed you in the chat log, I get killed by arcane sentries a lot (those purple laserbeams of doom)