Why did a Dalek beg River Song for mercy?

Question

In the season 5 finale, after a Dalek shot The Doctor,

DALEK: You will be exterminated!
RIVER: Not yet. Your systems are still restoring, which means your shield density is compromised. One Alpha Mezon burst through your eyestalk would kill you stone dead.
DALEK: Records indicate you will show mercy. You are an associate of the Doctor's.
RIVER: I'm River Song. Check your records again.
DALEK: Mercy.
RIVER: Say it again.
DALEK: Mercy!
RIVER: One more time.
DALEK: Mercy!

Why would a Dalek cry for mercy when they are stripped of emotions?

Answer 1

The Dalek article on the Doctor Who wiki says:

The emotional capacity of Daleks was limited to largely negative emotions (hate being chief among them), as "human" emotions such as compassion and sentimentality were considered by them to be weaknesses. However, they were capable of comprehending these emotions, and often used them to exploit their enemies.

I'm not sure whether this idea was established in the classic series or if it's a new idea created by the revived post-2005 series. For example, in Genesis of the Daleks (1975), when the Dalek's creator Davros gave instructions to a scientist on the genetic modifications he wanted, the scientist objected "It'll mean creatures without conscience, no sense of right or wrong, no pity. They'll be without feeling or emotion", and Davros answered "Correct." So that seems to suggest that in this conception, they were entirely lacking in emotion. But in the 2005 episode Dalek, the Doctor said "The Dalek race was genetically engineered. Every single emotion was removed except hate." Similarly in the recent episode Into the Dalek (2014), the Doctor said "Dalek mutants are born hating. This is what stokes the fire, extinguishes even the tiniest glimmer of kindness or compassion."

I don't know of any dialogue that established they can feel fear or other "negative" emotions other than hate--the closest I could find was The Parting of the Ways (2005), where the Doctor said "Do you know what they call me in the ancient legends of the Dalek Homeworld? The Oncoming Storm. You might've removed all your emotions but I reckon right down deep in your DNA, there's one little spark left, and that's fear." That seems more like a speculation that something he knew for sure though. And as the wiki quote above says, they can at least understand human emotions well enough to try to manipulate us with them, and it's possible that's all the Dalek was doing when it yelled "mercy!"

Answer 2

Well, S9 Episode02 may answer your question: "Mercy" is the feeling Daleks legacy from Davros and Davros got the feeling in his childhood from the 12th Doctor.

Answer 3

Daleks, are not stripped entirely of emotion, merely of what we would term positive and constructive emotions, such as compassion, love, and, of course, mercy. The Dalek still desires, strongly, to continue to exist, and knows that mercy characterizes the Doctor, so it uses that to try to sway River. No (positive) emotions required.

Answer 4

Another thing to remember - These are Not Quite Proper Daleks

In this weird little alternate version of Earth, penguins lived on the Nile instead of cats, Polar Bears live in the desert, and only madmen see stars.

As a display in the museum, they didn't even quite know what it was - like the rest of the anomalies, it's explained with "they shouldn't exist, and yet they do"

So in addition to just rebooting after being exposed to a little bit of the old universe, they're Daleks from a different, smaller universe, so how those Daleks act and respond may be slightly different as well.