Does "asking" as a noun have much, say, historic use?

Solution 1:

Þære spræce and þære ácsunge

You suggested perhaps the 1800s for the origin of this use: I’ll see that bet and raise you a millennium. Not an Americanism but perhaps a Wessexism, for some one thousand one hundred and thirty years ago in ᴀᴅ 885, King Ælfred the Great wrote not only:

Hit is þeaw þære spræce and þære ácsunge.

but also:

Mid ascunga.

Much more recently, Shakespeare wrote in King Henry VIII:

Bestowing on him, at his asking, The Archbishopricke of Toledo.

And again in Coriolanus:

Yet dare I neuer Deny your asking.

When Tennyson wrote Tithonus, he deliberately used this archaizing word to lend an air of antiquity to his verse:

Then didst thou grant mine asking with a smile.

This deliberately hearkens back to olden times, where such things as this from the 1410 Pynson translation of Bonaventura’s Myrrour were readily found:

Our lady··answereth sadly and shortly to theyr askynges.

Almost all such uses of asking save for Ælfred’s are now considered archaic or obsolete by the OED. The King’s sense of the word was “the action of putting a question, interrogation, inquiry”, and this is the one still employed in yours for the asking.

Inquiries quickly become requests, and Shakespeare’s Henry VIII use (“the action of requesting a favour, gift, etc.; praying, begging”) is also still considered current. However, his Coriolanus use (“a petition, prayer, a supplication”) is now archaic.

There is also the regional use of an asking meaning the publication of marriage “banns” (proclamations of marriage), but this is very limited in distribution.

Summary

Even though askings as a count noun has existed since time immemorial, it is now sufficiently obsolete that most spellchecking programs will flag it as an error in Modern English.

To most people, “What’s your ask?” and “Did he send in his asking yet?” sound like nothing but annoying business jargon instead of like forgotten pieces of English dug up from some ancient graveyard.

_{All citations are from the OED.}