This general construction is commonly used:
I gave <person> <reward> to <task>. 'I gave Billy five dollars to wash my car.'
That's not quite the context we have here, where <reward> is given in advance as an inducement to be quiet, but the construction seems to me to be fine just the same. 'I gave Billy five dollars to vote for me.'
Yes, that's OK
I disagree about the "kid" thing (I hear it all the time—my kids are at the pool, my kids are great at hockey, etc.) but I wonder if maybe changing the verb would help. I bribed my kid with a sweet to be quiet.