Learn English – Etymology of the idiom “by and large”

In the Wiktionary scutwork entry, one of the references, Medical meanings: a glossary of word origins, suggests either scuttle, or OE scitan (excrement) as possible origins of the scut in scutwork. I think scitan is plausible; note from the etymology of shit:

Noun use for "obnoxious person" is since at least 1508

Wiktionary also suggests for scut as "contemptible person" that the etymology may be from the obsolete sense of scout, which Etymonline explains:

"to reject with scorn," c.1600, of Scandinavian origin (cf. O.N. skuta, skute "taunt"), probably from a source related to shout.

I think you're probably right that meaning (2) of scut is more closely related to scutwork, but note that calling someone contemptible doesn't have to be unrelated to meaning (1) as in the quotations in Wiktionary, or, for example, this 1907 story:

"Don't understand me?" roared the farmer. "What have you done to my darter, you scut, sitting there preening of yersel, like a silly peacock in the sun?"
The angry man approached the bed swiftly, and, seizing Aubrey's pyjama-clad shoulder in a massive fist, thrust his hairy face within an inch of his victim's nose.
"What have you been doin' to my lass?" he roared again. "Now d'yew onderstand me?"
"W-what has she been telling you?" Aubrey was horribly frightened.
"Summat that made me sware you shouldn't leave this house till you'm man and wife all raight and praper accordin' to law."

An answer at answers.com quotes American Heritage Dictionary of Idioms to the effect that to not give someone the time of day means

Ignore someone, refuse to pay the slightest attention to someone, as in He's tried to be friendly but she won't give him the time of day. This expression, first recorded in 1864, alludes to refusing even to answer the question, "What time is it?"

By contrast, an answer at tywkiwdbi.blogspot.com says:

the expression goes far back beyond the time when people wore watches... In Shakespeare's day, the meaning was quite clear. "Good time of day" or "fair time of day" was a salutation just like "good morning" or "good evening"...We no longer greet people by saying "good time of day," but we still use the idea of giving such a greeting as a sign of favorable attention. In other words, refusing to give someone the time of day is thinking so little of him that you would not say hello to him on the street." (credit to Scribal Terror)

If tywkiwdbi and Scribal Terror are right, then the expression is rather older than the AHD Dictionary of Idioms said.

Edit: The BBC Learning English presentation “Not give someone the time of day” says:

Long ago, in Shakespeare's time, the phrase 'good time of day' was a greeting often used. These days we say 'good morning' [or 'good afternoon'] … So to say that you wouldn't give someone the time of day means you wouldn't want to greet them or say hello. So the saying means you refuse to give someone your attention.

As an example of the phrase used in Shakespeare's time, consider King Henry VI, part II, Act III, scene I, as Queen Margaret speaks:

Can you not see? or will ye not observe
The strangeness of his alter'd countenance?
...
We know the time since he was mild and affable,
...
But meet him now, and, be it in the morn,
When every one will give the time of day,
He knits his brow and shows an angry eye,
And passeth by with stiff unbowed knee,

Edit 2: Another example from Shakespeare (as pointed out by ΜετάEd's reference) is from King Richard III, Act I, scene III, when Buckingham says “Good time of day unto your royal grace!”. Note, Shakespeare is believed to have written both plays about 1591; he might or might not have put 16th-century speech into the mouths of 15th-century people. Henry VI was King of England and/or France at various times between 1422 and 1471, and Richard III from 1483 until his demise in 1485.