What is the chance of the AI granting the Lotharingian crown to Burgundy when you request it? I am curious 
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It depends on the AI event choice settings. It's option B of this Austrian event, so it could be 5%, 50%, or 0%.
There are multiple versions of this event from what i've seen. If you have >170 relations with austria they will grant it.
Since the Burgundian duke can also be crowned King of France, it would be great if there were more events that could eventually lead to the unification of the three kingdoms that comprised the Carolingian Empire. Maybe through becoming King of France, Lotharingia and getting elected Holy Roman Emperor.
Well, event-wise, if he doesn't get the Lotharingian Crown he can, though you can question why that'd stop it completely.
How can you request the Lotharingian crown? Is that an event? Or do you need to occupy some lands?
There are three events with restrictive triggers where Burgundy can request the crown from the Emperor.
It´s BUR 28004, BUR 28007 and BUR 137049 and yes you need to own 7 or 8 provinces to even have a chance to get those events.
The events are located in this file if you want to look what the exact conditions are:
AGCEEP_Specific_Burgundy.eue