Something has always bothered me about the first movie, I'd enjoy hearing your comments and ideas.
Vito was the richest, wisest, most powerful of all the Dons, yet he left himself wide open on the Solotso deal. Why did he send Luca off to investigate the situation with such a lame excuse, telling Sollozzo and Tattaglia that he wasn't too happy with the Coleone family. Considering how loyal and trusted Luca was, this just seemed stupid. If Vito was so smart, why didn't he see this coming?

As the Turk said, "The old Don was slippin'". He underestimated the craftiness of the opposition, I think.

Don C. was smart enough to think that Sollozzo might have something up his sleeve. That's why he sent Luca on the mission. He just wasn't smart enough to realize that Sollozzo would see through it.

Yes, indeed, he was slippin'. Of course, it also served a dramatic purpose: It set the stage for everything else that happened in GF.
A major theme in the Trilogy is how people underestimate each other, leading to disastrous consequences. The Don underestimated Sollozzo. Sollozzo and McCluskey underestimated Michael. Sonny underestimated Carlo. Carlo and Barzini underestimated Michael. Michael and Roth underestimated each other several times. Michael underestimated Altobello. Altobello underestimated Vincent. Etc., etc. As the Don said in GF: "Women and children can afford to be careless. Men cannot afford to be careless." At that point, he was speaking from sad personal experience.