I like where you are going with this plot idea. It sounds really awesome!!
As far as your questions go:
1. Yes, Lucrecio does have a claim to the throne. The man who took the empire from his father clearly failed as a leader, allowing himself to be enthralled by the Supreme Archbishop; clearly leaving an opening he could exploit. The only issue that would block his possible ascension is if people in power made a large fuss about his whole Patricide issue...
2. Because of that Patricide issue, the Church would probably oppose him on a moral front, but were he able to promise (and maintain) stability and a return to order, they would most likely side with him. The only problem there is what ever he has been doing with his Crows and the Wissenschaft society that he has been so careful to keep hidden.
3. If you go by the timeline, the Giovanni Emperors lasted until 957, with Lucrecio killing his father as a "child". The present (game wise) is the year 989, so most likely Lucrecio is between 41 and 47 years old. This is supposing he killed his father between the ages of 9 and 15. But in reality you can make him as old or as young as you want, it is your story after all.
4. A political marriage is possible, although fairly unlikely if you look at the geneology. Elisabetta is the daughter of Emperor Elias Barbados, who was the "political"(I have no idea what they mean by that) cousin of Emperor Lascar Giovanni, father of Lucrecio. By our standards that would mean they would be First Cousins. Marriage is not unheard-of between cousins within Aristocratic lines, but it does generally disgust the public as well as lead to genetic defects within the family-line. Don't be discouraged by this though, again, it is YOUR STORY, so you can omit or include as many (or as few) details as you like.
As far as the future of Gaia following a possible union, who knows? Maybe another age of prosperity and returned power as the Empire reassurts itself upon the disobedient lands. Maybe the revolting lands return to the fold with the "liberal" leadership of Lucrecio to look forward to. I would love to hear your ideas about what would happen following such an event.
Thanks for reading the rant...