The word used in the situations you described is (transliterated) yada, which simply means "to know" and is often translated (in different circumstances) find, understand, comprehend and acknowledge.
Considering some of the situations presented in the Old Testament (drunken incest of Lot and his daughters, rape of Dinah, rather suggestive imagery in Song of Solomon, etc), I doubt euphemisms would even be considered necessary; however, Strong's concordance suggests it is used euphemistically (in addition to other senses) in the Hebrew.
I imagine those who translated earlier (KJV) would not have seen reason to translate yada any differently when it referred to sex (they rarely translated words differently solely for the sake of clarity), whereas newer translations (NASB), more for the sake of clarity than anything, use another euphemism (ie. had relations with). I doubt any translators would find a reason to change a euphemism in the original language into something more graphic.
Deosil comes from Gaelic (both the Irish and Scottish forms, found as deiseil, deiseal and deasal), and means "right", being the direction one turns when going clockwise.
The spelling deosil though is rarely seen prior to Gerard Gardner, and is likely just his choice of how to spell a dialect word that had no set spelling (the OED lists deasil, deiseal, deisal & deisul). It holds the distinction of being the sole example of several "witch words" that he claimed were "seemingly Celtic" or from "some older tongue" that weren't in fact plain (if obscure) English words of Saxon origin. Gardner ne cuþe a Sax word gif it bat him in þe ærs (I'm probably getting that Anglo-Saxon very wrong, but I'm still better than Gardner).
Anyway, it still exists in Irish and Scottish Gaelic as a word meaning right (as in not left), right (as in not wrong), south (which is on your right as you face the direction of sunrise) and as a general exclaimed blessing ("may things go right").
It has nothing to do with the direction of the sun, as is often suggested (with people even suggesting that it means anticlockwise when one is sufficiently south of the equator).
Withershins is in fact a merging of two separate words, with overlapping meanings. The most direct ancestor would be widdersyns (and similar other spellings) which as you say (and show from etymonline) means "against the way" and hence contrary to the "proper" direction.
However, there was also once widersonnis, withersones, etc. where the second part of the word comes from sonne, sunne (sun) rather than sinnen (way, journey, direction), as with:
Sayand the said Margarat Baffour vas ane huyr and ane wyche and that sche ȝeid widersonnis about mennis hous sark alane.
(Claimed the aforementioned Margaret Balfour was a whore and a witch and went [in the direction contrary to that of the sun] around men's houses [wearing only a light nightshirt or shift which is a shocking state of undress to the people who spoke Middle Lowland Scots, and so they apparently needed a word for it, though its a right bugger to translate into Contemporary Modern English]).
Of course, the two words were close in both sound and meaning, and in some cases when we have forms like widdershines it's not even clear which it is. As such, the two formerly separate words merged into each other, and so we have one of the interesting cases where a word as two separate (albeit similar) etymological roots.
The pairing of the two is another etymological case in itself. Deosil is Scottish Gaelic (as well as being Irish) and as such predominately found in the Highlands. Widdershins is Scots word, predominately found in the Lowlands. This oversimplifies a lot, and even from that oversimplification, there is nothing to say that a borrowing wouldn't have led to both being used by the same people (I grew up with a dialect with borrowings from both Irish and Ulster-Scots, so I certainly won't claim it doesn't happen), but just how likely one was to find the two paired is unclear. A lot of the written records we have of both words are from English writers who are pointing out the words new to them as matters of interest in themselves, as they travelled in Scotland, which muddies the waters further.
Whether the two were much found together or not in 16-19th Century Scotland, as a pairing used in Contemporary Modern English, they originate with the Wicca and other Mesopagan and Neopagan practices. So too does the idea that they are "the old way" of talking about clockwise and counter-clockwise; in truth the old words for those rotational directions were right and left.
Best Answer
It has its origins in Western European etiquette, basically the rules of conduct in society.
In a number of situations, it was (and largely still is!) deemed correct behaviour to give ladies precedence over gentlemen. This is for instance the case when serving food or drinks at a table: one servers the ladies, then the other guests, then the host.
It is interesting to note that the common use of ladies first actually often contradicts traditional etiquette:
The phrase is often used to mean after you, but etiquette does not prescribe that ladies should always pass first through a door or other narrow passage. The idea is that the gentleman who accompanies the lady will only let her pass first when entering a trusted environment, such as his or her own home. When entering an unknown place, or when exiting a building, the man is to go first. The reasoning behind this is that if any danger lurks on the other side of the door, the strong man can defend the poor defenceless damsel.
It is also traditionally considered bad form to invite a lady to "go first" when climbing stairs - a situation in which the gentlemen might indecently stare at certain parts of the female anatomy as she moves up and ahead of him...
When in a movie (or in real life) one man tells another "ladies first" instead of "after you", it is a joking insinuation that the other person performs the female role in the relationship between the two men - the speaker is asserting his leadership in the situation.
disclaimer: any references made to women as the weaker sex in need of protection are made solely as historical observation and do not necessarily reflect the personal opinions of the author of this answer!