Continuity mistake: When they jump in the pool, they drop the guitar. When they jump out their guitar is no longer in the pool.
Other mistake: When Miguel is singing "Un poco loco" on stage, he goes out of range of the mic, but his voice is amplified.
Continuity mistake: Miguel is thrown into the pit and lands in water, being completely submerged. But a couple minutes later when Hector mentions Coco, Miguel pulls the family picture out of his pocket, and the picture is totally dry and showing no effects of being submerged.
Continuity mistake: When Mama Imelda is singing on the stage and in the shot where she runs towards Miguel and Héctor, you can see several security guards on the stage attempting to apprehend her. However in the next shot that shows Ernesto grabbing her hand, there are no security guards visible. Three of them reappear briefly (the shot lasts about a second but you can see them) but they disappear for the rest of the scene. (01:23:05)
Continuity mistake: When Miguel is using a bottle to make some music the fruits in the background change between shots.
Continuity mistake: When Hector confront Ernesto just before he gives Miguel his blessing, he peels off the Frida disguise. But his beard on his chin is missing. It shows up in the next scene. (01:04:23)
Other mistake: Miguel was drowning in the swimming pool in De La Cruz's party but when he was thrown in the pit he managed to swim out of it.
Answer: It is never actually addressed what Miguel told his family about his adventure, so it can't be said whether or not they believe him. It would seem to make sense that he would tell them though, it was a pretty fantastic tale after all and I'm sure he would want to tell his family about it. It would also make sense that his family would believe his story, it's really the only way he could have known that Remember Me was a lullaby Hector used to sing to Coco. The letters Coco saved prove that Hector was the actual writer of De la Cruz's songs and it was later proven to the general public, hence the graffiti on De la Cruz's memorial and a memorial for Hector taking it's rightful place in town. It can also be reasonably assumed that the general public is aware that De la Cruz murdered Hector since the theft of the music and Hector's death go hand in hand. It wouldn't make much sense for people to hate De la Cruz so much if all they thought he did was take credit for his partner's music after his partner was already dead.
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