Well yeah, it's hardly unusual that no one has a clue what's going on with partial-registrations in South America. Just look at the drama it was getting Mascherano to Anfield. Ditto Tevez.
In Di Stefano's day there was a player's strike in Argentina. Some of the players left for other countries to earn a living - Colombia in his case. When the strike was settled there was a lot of confusion over whether the original Argentinian clubs or their new clubs owned the registrations.
The Franco conspiracy largely goes that the Spanish FA suggested something so ridiculous --him playing for both-- knowing Barcelona would walk away. Seems a bit of a reach to me. Football associations and incompetence are never far away from each other.
Franco himself was almost totally uninterested in football. Though a strong Barcelona would obviously not be in the state's interest.
Franco favoured Real Madrid a lot in the 50s and 60s but it's a myth that Real Madrid were Franco's club. People see Madrid now as the city of the establishment, while Barcelona are rebellious and against the rest of Spain, but Madrid was actually one of the last places that Franco's army couldn't take during the civil war and the people of Madrid fought so hard that he even considered changing the capital city when he won.
As you said, he had no interest in football but he believed that having a strong team from Spain would improve their national image so wanted to create a powerhouse of European football. The new European cup was, to him, a perfect opportunity to showcase his country's footballing talent to the Continent.
However, when he first took power, the most successful team in the country were Barcelona and he actually did favour them a lot in the early years in the same way he later did for Real Madrid. Real Madrid emerged at pretty much the perfect time just before the first ever European cup and by this time, Franco had switched his efforts to them and they won the first 5. Of course, it was all very corrupt the way that happened but Franco definitely didn't support Real Madrid. He didn't care about football.
