I'm going to pass on the "should" question. My instinct tells me it could be ambiguous, because "should" has several uses. If you use "should" in the sense of "ought to," then it would be redundant. You'd be saying it's necessary that they ought to do it. And, in fact, you'd no longer be saying that they
should do it. You'd be making a major change in the meaning. I hope I'm making this clear. We're talking about two things, the
obligation to be honest, and the
act of being honest. If you insert "should," you're now saying that the
obligation is necessary, and you're no longer saying that the
act is necessary.
But I have this feeling there's another sense of "should" which would be permissible - like "if it
should happen to rain." (Maybe I'm starting to lose that feeling.)
At this point, I have to advise against it. You may shorten your sentence to "Children should be honest all the time," and it would mean about the same thing, although I believe "necessary" suggests a stronger obligation than "should," perhaps approaching compulsion.
On the pronunciation question, it's "thuh same." We use "thee" only before (non-silent) vowels, like "thee apple," or when the vowel is preceded by a silent consonant, as in "thee honest man."