You'll notice that these kids are singing with microphones: by definition, unless the composer specifically calls for amplification (I'm looking at you, John Coolidge Adams), it's an unamplified art form. Kids do not, as a rule, have the vocal power to be heard over an orchestra without an electronic boost. And Puccini's orchestration tends to be fairly heavy. Cio-Cio-San is one of the longest and most difficult roles in the soprano repertory!
I linked in previous comments to Renata Scotto's performance of "O, mio babbino caro" in a Metropolitan Opera Gianni Schicchi, because she is so very stylish and because her
But anyway. What age is too young for opera?
Here's a long list of works in the standard repertory that call for children, usually in the form of a chorus. The music written for children's choruses is, above all, appropriate for their voices and age. If your child likes to sing and has been in a chorus, and you can deal with the rehearsal and performance requirements (the children's chorus in Die Frau ohne Schatten is, I think, in Act I and in the last ten minutes of a very long opera; there will be seated rehearsals, separate chorus rehearsals, and staging rehearsals), by all means, have the child audition for your local opera company. The Met's children's chorus requirements are here and they are rather time-intensive; your local, lower-budget company will vary.
What about solo singing roles for children? Well, the longest and most demanding has got to be the boy soprano role of Miles in Britten's The Turn of the Screw, one of the greatest of 20th c. operas. I have also seen a production where his sister Flora was taken by a girl, but I think it is more common to have a small adult soprano in that role. I note that the first singer I heard in this role was Michael Kepler Meo, in Los Angeles in early 2011. He took some time off the stage and returned as a teen-aged tenor for The Secret Garden earlier this year. The role he played in that opera was written with him in mind, meaning it took into account his specific abilities, vocal range, power, etc.
There are short children's solos in other operas, such as the shepherd's song at the start of Act III of Tosca; I saw the role of Amore sung by a boy soprano in SF Opera's L'incoronazione di Poppea back in 1998. There is the non-singing role of Dolor (sadness, trouble) in Madama Butterfly.
Okay, let's get to the real point: when should a child start singing opera solos written for adults? Well, it depends on the kid and his or her voice. The great Spanish mezzo Conchita Supervia seems to have her sung her first Octavian (Rosenkavalier) at the tender age of 15. About 12 years ago, I saw a completely professional Cherubino (Nozze di Figaro) who was around 14 or 15; this was in a smallish house, the 900-seat Lesher Center in Walnut Creek.
Dame Eva Turner demanded singing lessons, and got them, at age 10. At 14, she sang "Ozean, du Ungeheurer," from Weber's Oberon, in a recital, presumably with piano. (It was another decade before she sang complete operatic roles on stage, however.) Licia Albanese very likely made her professional debut at the age of 17 or 18.
But for the most part, there's a progression from lighter music, such as 18th c. Italian art songs or Mozart arias, to heavier music. Most singers don't do 19th c. operatic music until they are around college age. Voices develop at different rates, because no two bodies are the same. Frida Leider was singing Bruennhilde in her 20s; Ljuba Welitsch sang Salome in her 20s. Other singers are past 40 before they touch those dramatic soprano roles.
By and large, Puccini is big-girl music. A few years ago, I heard a college-age singer perform a light aria at a short master class given by Will Crutchfield. I remember thinking, whoa, Puccini voice! why isn't she singing "Un bel di" or something like that? I was gratified to hear Will say something similar to her. In any event, it's not for kids.