I believe this is a variant of "hello the camp", which is used as your father describes: when you're about to enter someone's encampment, as a kind of warning/greeting in lieu of knocking on the nonexistent door. (There's an elided "to" in there: the full/grammatical form would be "hello to the camp".)
I imagine "hello the fire" could be used if there isn't an encampment to speak of, just a fire with guys sitting around it.
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:
Between 1880-4 I spent a considerable time in Constantinople and Asia Minor, where I played what was then called 'Biritch or Russian Whist'. I was then living, while in England, at Cromwell Road and introduced the game to many of my English friends, who liked it so much that they asked me to have the rules printed. ... 'Biritch' was attributed to the Russian colony at Constantinople; in my time the dominating social and political element. [not my ellipses, but the website's]
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.
Best Answer
I read "fire away" almost the same way that I would "bombs away", as an order to "release" the projectile, not a suggestion for where they can put their projectiles. A longer form might have been "let the fire be away" or "let the fire be on its way". This has the feeling of something that could be an actual order as opposed to an order to whiff, which militaries tend not to do for reasons involving accidental murdering.