1. By definition this has to occur sometime between 1867 (the Ausgleich and creation of AH) and 1885. Sorry but I just don't see it happening. As has been stated, Franz Joseph had better things to do the worry about sending various people who could barely get along with each other at home to some God forsaken jungle in the middle of Africa. Rudolf isn't the answer either (or for that matter Maximilian). Judging solely by Rudolf's beliefs on how the Empire should have been liberalized and reformed I can't see him being terribly big on the whole Imperialism thing. Any chance of real Habsburg colonization efforts died with the Ostend Company. Perhaps an attempt could be launched under Franz II, but realistically after that... no. Sorry.
Rudolf's liiberalism would not necessarily stop him supporting a colonial venture. In the late 19C many men of progressive views supported imperialism as part of the "march of progress" bringing enlighenment to the backward parts of the world. Only in the 20C did it come to be seen as a specifically right-wing policy.
As for Max, well, if he thought that being ruled by a Habsburg would benefit the Mexicans, why not the Congolese?
2. If I ever hear anyone say Franz Joseph needs to go in reference to removing him for power for one reason or another again, I am going to slap the speaker so hard they'll be seeing the jewels on the Crown of Rudolph II for weeks.
Trouble is, this is difficult to avoid in any Austrian WI.
FJ was averse to change, and generally accepted it only under duress, so it usually came too late. If you want any change to come "in time" you are virtually obliged to remove him. Whether this would save the Empire is of course another question. Well meaning liberals like Max or Rudolf might just have made the roof fall in that much quicker.
Incidentally, there was an attempt on FJ's life in 1882 - google "Oberdank".