Basically you are talking of a Blitzcrank except instead of grabbing Monsters/Champions it grabs projectiles, reflecting them, as well as a Metronome-Type champion that at the click of a button, possible a 2-second cast time, shoots out a random ability from any champion. An ulti where if someone ults they can do it again, basically give everyone the Darius advantage. Global Pull/Stun I can't think of a champ like that but it sounds a little far fetched. Here's my opinion.
A champion can be made like this but at first he would be extremely powerful. Do not be surprised if he was nerfed at some point. I think if the Champion had the ability to allow an ally to use their Ultimate again it should come at a price. Perhaps instead of an ability it should be an item specific to that champion. Call it the "Reverse Blade" and when it is cast on an Ally it automatically resets their ultimate ability. It would not be a strong item, maybe a 3000g Total Cost, and it gives mostly Health and Armor. A Global Pull/Stun is very over-powered and is not a good idea. They do have Global Abilities such as Karthus but it is not a Pull or a Stun, simply a Global Burst. So I believe that a global pull or stun is out of the equation and would be hard to pull off. Perhaps a Global Stun would be feasible but it would be simply that, a stun. It most likely should not do very much if any damage. And all the abilities at random is a very cool idea, but think about how hard it would be to create such a champion for one, and for two, the roll table for the moves would be incredibly bad and in team fights you would have to get lucky to be useful. I think it should most likely be his ultimate if anything, and it should look something like this:
R - Metronome (60-Second Cooldown)
"(Champion) rolls a dice and as it rolls across the floor it lands on a number, casting a random Ability from a champion of that number, the Ability will not be cast automatically, but (Champion) will have 3 seconds to cast it before it disappears and Metronome goes on cooldown."
The ability should be able to include any ability EXCLUDING ultimates. Meaning they can get anywhere from Sona's Heal to Kha'Zix's Leap, but no one's ultimate. Thus making you have 4 Basic Abilities but no ultimate whatsoever, which in certain situations could be rather helpful.
I think that your idea is slightly flawed but actually could be feasible with a few tweaks, and definitely makes me think about what could appear in the future!