What if Italy had joined with Russia against the Ottomans?
Would the Ottomans lose earlier or around the same time (or would Italy joining against them ensures they win..)
I don't think it would necessarily make a huge difference to the central theater of the war effort in the South Eastern Balkans, since Italian operations will be concentrated in North Africa which was far less important from the Ottoman perspective. The biggest potential impact is probably an indirect one- that the Greeks will enter(since the Italians will have aspirations on Crete and the Dodecanese, and the best way for the Greeks to secure them for Greece is to enter the war themselves). The Greek entrance might hasten the Ottoman surrender a little, but it's primary consequence will be the possibility of even greater demolishment of the OE at the treaty of Berlin then OTL. They might well lose everything in Europe except their Thracian territories.
Though the Ottomans might put up a stronger fight then you would think. They had a larger navy(my recollection is that it was the third largest navy at the time) then Italy, though I assume it was of lower quality, and they had forces staged on the Greek border... and OTL the later Italian invasion of Libya struggled to put it lightly.
And of course Britain might not take kindly to the complete demolition of Ottoman territories and increased in Italian power in the Med... they won't directly enter the war, but they might use diplomatic pressure to effect a ceasefire earlier then in OTL.
What could the Italian Army manage to take, and what parts of the Ottoman Empire could end up Italian?
Tunisia, Tripolitania and maybe Cyrenaica. Crete and the Dodecanese are possible if Greece doesn't enter war/gets screwed at Berlin. The Ottoman's Ishkodra province(roughly the northern two thirds of Albania) being in the Italian sphere of influence is also possible/likely if my scenario of near total demolishment of Ottoman territory in Europe occurs.
Would Italian Immigration focus towards these new conquests instead of the Americas and what affects would that have?
That would be state policy- whether it works out that way is another matter entirely, since Italians proved reluctant to move to Libya decades later... certainly a very large number did, but not as much as the state wanted. Certainly the Italian population in North Africa will be larger then the OTL colonial population, since a lot of Italians immigrated to America or Algeria before 1911 when Libya was captured, and some of that three decades worth of immigrants will surely end up immigrating to Libya and Tunisia instead. Probably a large fraction then OTL as well since Tunisia has a nicer environment and more civilized culture then Libya(at least that's my understanding).
Since the Italian colonial population came close to a majority IOTL, it probably passes that point with this alternate scenario. Of course even if it reaches a majority that doesn't mean it stays a majority, given the birthrate of marginalized peoples tends to be higher then conquering population. Unless of course the Italians go for all out genocide(rather then just the callous mass murder of OTL).