Hmm, it translates more literally to "I weigh (to) my son," rather than "at." Think of it in terms of: any action that involves or is done to a person requires an "a" after the verb. It's just how Spanish is: no language has to translate perfectly into English grammatical rules!
That's exactly how I answered! Seems to me it's more frequent to say "I kiss my son" than "I weigh him"!
This misunderstanding may comes from the Spanish "p" having no air behind it. Try this: light a match (with adult supervision, of course) and say "pee" without moving the flame. After a few repetitions you'll be able to say it almost like a "b" -- which in English has no air behind it either -- but distinguishing it as a "p" with the tiniest little pop. It's fun to try.
I thought the exact same thing. It's the way the speaker pronounces it. They really should switch up the speakers because there is a definite regional or country-specific sound to the way she says words; especially the "yo" as "jo" which I rarely have heard. It would be like listening to a spanish speaker from spain with the lisp (which I honestly am more used to), for those not accustomed to it.
It's necessary in Spanish. Here's how I think of it: any action requires an "a" after the verb if it involves or is done to a person. It's just how Spanish is. After you practice this requirement, it'll come more naturally to do. An example to consider: "Tengo que llevar a mi hijo al doctor." Espero que eso te ayude. (Ask me why I don't use "ayuda" in that sentence ...)