First of all, I think I'll make La reine africaine canonical, given that it was the French title of the film.
International Congo is definitely a mess. Some parts aren't as bad as others - I've mentioned that Bas-Congo is run by a neutral Spanish governor who's a strong ruler and strict with the rubber companies, and Dietmar Köhler, who's a competent administrator and includes Africans in the upper ranks of his government and army, isn't so bad if you can get past the fact that he's a ruthless warlord. But for the most part, the provincial governors and rubber companies are trying to fight their countries' enemies, carve out vest-pocket kingdoms for themselves, or both. There will also be a couple of African armies and would-be warlords in the mix, and some of them may have a touch of the apocalyptic.
The war in the Congo will involve small numbers of troops compared to the main African theaters, let alone the European ones, but it will be a nasty, give-no-quarter sort of war, with rubber, labor and land as the prize. And it will spill over into the neighboring countries - this won't be the only time the Anglo-Omani empire gets involved. Some of the Congo may well be administered de facto by the Anglo-Omanis, the Portuguese or the French by war's end.
The European powers won't hesitate to intervene after the war, whoever wins - given that the Congo is an international mandate rather than an independent state, they'll see themselves as having every right to do so. Reconstructing a functioning administration will be another story, though, and bringing some of the warlords to heel may be easier said than done. The Congo will occupy an inordinate amount of attention at the peace conference.
Tanganyika: As Falecius says, much of it is British in all but name at this point, and has been getting steadily more so for some time. That isn't the case everywhere, though - where there's a strong indigenous ruler who has won a place in the Omani nobility, or where the local feudalist is present rather than an absentee, there's more autonomy. The British have been good at creating and filling power vacuums, but they haven't interfered directly where strong local power structures exist.
The Sultan's army at this point is guarding the Ethiopian frontier in case Menelik II decides to jump into the war, and now that the Congo conflicts are starting to spill over, he's raising more regiments to protect his western provinces. If Ethiopia stays quiet, the Omani-Zanzibari empire may also become a secondary source of troops for other British theaters.
And Indian troops will certainly be appearing - all things in time.
Finally, a question for wolf_brother and Falecius: What would the Pope's likely reaction be to the seizure of Rome by an anti-clerical Italian state during a period of global warfare? Is there a chance of another Avignon papacy, or would he be more likely to retreat into the Vatican like Pius IX did in OTL?