The etymology of "versus" is pretty simple. It came from Latin and it was originally used in English in the law sense (as it's still used today: Roe v. Wade). Later on it started to be used more flexibly.
There are several abbreviations that have been used for versus: v., vs., and ver. Both vs. and v. (with and without the period) are mentioned in Oxford Dictionaries as currently acceptable abbreviations.
The Oxford English Dictionary has this as its first citation for versus:
Also the jugement by twene Broghton and the Glasier and by twene John Husset versus John Notte and specially of Sr John Notte of his fyn.
John Shillingford's letters and papers (1447–8)
(The MED has the same quote as its first citation for versus.)
It's pretty easy, at least for me, to see the similarities between the legal sense and the general (or sports) sense.
I also looked up contra and vel in the OED. Contra has never been used in English in quite this way (i.e. placed between two people or things to indicate opposition). Vel doesn't even have an entry (although it is part of the expression vel non "or not"), meaning that it was never really used in English.
Lastly, according to the answers given to the Latin SE questions “Us versus them” - opposite of “noster”? and What's the difference between vel, aut, -ve, et cetera? it would be aut (not vel) that would be used in Latin between two opposing parties.