Limit On Number Of Simultaneous Actions? [duplicate]

Alright so, as far as anyone can tell, there is no global cooldown in league. Just like in the answer to the question Colin D posted, almost every ability in the game has a cast timer as well as a cooldown. Some abilities have a short cooldown timer to prevent them from being spammed instantly (akali's ult). There are few exceptions to this, most being instant self cast abilities. Fiora's deflect and attack speed boost are just one example of this.

In regards to your issue of actives not being activated, I recommend smartcasting your item bar. I have my '2' key smartcasted for whenever I have BOTRK because its a lot easier to press 2 in the general area of an enemy instead of trying to focus on trying to click activate it on them in the heat of battle while trying to use abilities at the same time.

To answer your main question, there is no (documented) limit to how many abilities you can do at once. Your only limitations are cast times and cooldowns. You can remove a lot of these errors by learning to smartcast if you haven't already done so.