I've always thought that the name of this card game comes from the English word bridge (the structure) but it is not quite like that. It's the English pronunciation of a game called Biritch, which was also known as Russian Whist.
Etymology of the word from Etymonline:
card game, 1886 (perhaps as early as 1843), an alteration of biritch, but the source and meaning of that are obscure. "Probably of Levantine origin, since some form of the game appears to have been long known in the Near East" [OED]. One guess is that it represents Turkish *bir-üç "one-three," because one hand is exposed and three are concealed. The game also was known early as Russian whist (attested in English from 1839).
Furthermore from the book "The Theory of Gambling and Statistical Logic" By Richard A. Epstein:
Question
This is the furthest I can go in my research (looks like an answer already) but is it possible to go further? Even it says obscure, are there any sources that goes deeper and gives more details (for the following questions especially)?
Related questions:
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Why is the word corrupted to a word — bridge — that is already used? For
example, why not britch? Is there a folk etymology here? -
How come there is a Levantine and Turkish origin? Can it be that
Russians learned this game from Ottomans in the era of wars, or did they adopt a name from the Ottoman Turkish language? (Also, the passage from the source book mentions that Biritch is not a recognized Russian word.)
Note: "Biritch" does not sound like a mispronunciation of some Turkish word or phrase meaning no trump. Interestingly, a Turkish language etymology website says that the name of this game briç is a loan word from English bridge. But this is about the modern Turkish language; there is also the Ottoman Turkish language used at that time which I can't trace back.
Best Answer
Some of your questions are answered by this website, which contains a transcription of the original pamphlet describing biritch. Collison (the author of the pamphlet, and a railroad engineer who worked for a time in Turkey) apparently wrote a letter to The Saturday Review dated 28 May 1906, where he describes the history of the game. I quote:
There were many variations of whist played in Russia, which this game was similar to. Mari-Lou in the comments has found a source that showing biritch is a variation of an earlier Russian game called yeralash. So while it's not clear whether the word biritch was originally Russian, most of the rules of the game are.
The word "biritch" means (in the game) no trump, although it is unclear whether this meaning is connected to its etymology. Maybe somebody who knows Turkish could tell us whether biritch might be a Russian mispronunciation of some word or phrase meaning "no trump".
More information probably can be found in the original version of The Saturday Review letter and also in another reference given on the above website: Thierry Depaulis and Jac Fuchs, "First Steps of Bridge in the West: Collinson's 'Biritch'", The Playing-Card, Vol. 32, no. 2, Sep.-Oct. 2003, pp. 67-76. Unfortunately, I can find neither of these online.