Even Bismarck was unable to keep together Austria, Russia and Germany.
The first Drei Kaiser Bund was signed in 1871, and was already dead in 1875 after the Bosnian riots (Russo-Austrian disagreement obviously). The revival of the alliance came in 1881, and was apparently a Bismarckian coup, since Russia was upset by the outcome of the Berlin Conference of 1878. This time around it lasted a bit longer (1881-1887, with a renewal in 1884) but it eventually died since the Austro-Russian conflict of interest in the Balkans could not be resolved (and possibly did not have a solution).
As a comparison, the Holy Alliance (Russia, Austria and Prussia) lasted much longer, from the Congress of Troppau in 1820 (when Metternich successfully wooed Czar Alexander) to the Russian intervention in Hungary in 1849 - almost 30 years. It was certainly much easier to keep the unity when the Balkan sore was not yet open (although it was starting to fester: the Greek insurrection of the 1820s - branded as a reckless and criminal act by the three eastern Powers - and the refusal of Metternich to participate in the multi-national fleet sent to Greece waters was not well received by Russia). However the Punctuation of Olmutz (1850) saw Austria bully Prussia out of the Erfurt Union (an obvious humiliation for Prussia) and the war of Crimea convinced Russia that Austria was not a reliable ally.
In the end Bismarck had to choose an ally, and favored Austria over Russia (certainly not out of misplaced ethnic sympathy but rather because Austria could never be a serious rival for German ambitions, while Russia was huge, populous and a potential rival in a more or less distant future).
It is true that in an ATL where Bismarck chooses Russia, Germany can better influence the rate of Russian development but at a same time a Russia which becomes disenchanted with the benefits of the alliance and chooses to terminate it at some point might be a big problem with no easy solution.
From the POV of the French government Russia is certainly the best possible ally, if not tied into the German alliance system (Great Britain would be an even better choice but there is no way I see for a Franco-British alliance after the Congress of Berlin and 1900. France and Great Britain will certainly cooperate on many issues, but the British government will not given France the boon of a defensive alliance aimed against Germany - IOTL even in July 1914 Grey never pronounced the magic words which might potentially stop the spiral of ultimatums, mobilizations and counter-mobilizations across Europe). Russia was willing to give this guarantee with the added bonus of lucrative French loans for the infrastructures that Russia desperately needed.
If Russia is allied with Germany, France has a problem: the potential partners are Austria and Italy, neither of them satisfactory under many aspects. Theoretically having both of them as allies would be better than having one or the other; I'm however convinced that the mistrust and bickering between Austria and Italy while they were both in the OTL Triple Alliance would prove even worse in this ATL (and from a financial POV loans to Austria and Italy would be less lucrative than loans to Russia).
An alliance between France and Germany is certainly an utopia: the war of 1870-71 left a lot of bad blood. The only possible scenario for such an alliance might be the survival of the 2nd Empire in a weakened form (say Napoleon III goes to Paris rather than following MacMahon to the defeat of Sedan, there is a quick and less painful peace treaty and Bismarck follows his instinct to prop up the Bonaparte regime rather than risk a republic on the German border). Very hard to manage, though.
Last consideration: the Ottomans' bugbear is Russia. If Russia is allied with Germany, it will be in the interest of France and Great Britain to prop them up to keep the Russian bear locked in the Black sea and it will be very difficult for Germany to enter as a commercial partner.